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Part III General Relativity

Lecture Notes

Abstract

These notes represent the material covered in the Part III lecture General Relativity. Alarge part of the mathematical background (mostly up to chapter 9) is based on the moreextended lecture notes by Harvey Reall [5] as well as Hawking & Ellis’s “The Large ScaleStructure of Space-Time” [3] and John Stewart’s “Advanced general relativity” [4]. Latersections on the “3+1” formalism of the Einstein equations and the Lagrangian formulationof general relativity have been inspired to quite some extent by Eric Gourgoulhon’s “3+1Formalism and Bases of Numerical Relativity” [2] and Eric Poisson’s lecture notes on“Advanced general relativity” [6]. Readers will find all these references valuable sourcesto explore topics discussed in this lecture in more detail. Primary purpose of the presentset of notes is to provide a verbatim description of the material covered in the Part IIIcourse on General Relativity. Indeed, they bear a high degree of resemblance to thematerial as presented on the black board in the lecture theatre.

For further reading on the topic of Einstein’s theory of general relativity, there existsa wealth of books more or less directly deidcated to the theory. An incomplete list ofbooks is given as follows.

• J. B. Hartle, “Gravity, An Introduction to Einstein’s General Relativity” .

• B. Schutz, “A first course in general relativity” .

• R. M. Wald, “General Relativity” .

• S. M. Carroll: “Spacetime and Geometry: An Introduction to General Relativity” ;cf. also [1] .

• L. Ryder, “Introduction to General Relativity” .

• C. W. Misner, K. S. Thorne & J. A. Wheeler, “Gravitation” .

• S. Weinberg, “Gravitation and Cosmology: Principles and Applications of the Gen-eral Theory of Relativity” .

Example sheets for this course will be available on the webpage

http://www.damtp.cam.ac.uk/user/examples/indexP3.html

Make sure you do not confuse these example sheets with those of the Part II course ofthe same name on http://www.damtp.cam.ac.uk/user/examples .

Note that this course does not cover (in any depth) the topics of Black Holes and Cos-mology which are the subject of other Part III Courses.

Cambridge, May 2014

Ulrich Sperhake

1

CONTENTS 2

Contents

1 The equivalence principle 41.1 Statement of the equivalence principle . . . . . . . . . . . . . . . . . . . . . . . 41.2 Bending of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Gravitational redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Curved spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Manifolds and tensors 92.1 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Curves and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Covectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Abstract index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8 The commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.9 Integral curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 The metric tensor 253.1 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Lorentzian signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Curves of extremal proper time . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Covariant derivative 334.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 The Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Physical laws in curved spacetime 425.1 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Energy momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Curvature 466.1 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 The Riemann tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.3 Parallel transport and curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.4 Symmetries of the Riemann tensor . . . . . . . . . . . . . . . . . . . . . . . . . 506.5 Geodesic deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.6 Curvature of the Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . . 526.7 Einstein’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.8 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

CONTENTS 3

7 Diffeomorphisms and Lie derivative 557.1 Maps between manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.2 Diffeomorphisms, Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8 Linearized Theory 638.1 The linearized Einstein eqs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638.2 Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648.3 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668.4 The field far from a source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678.5 Energy in gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.6 The quadrupole formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9 Differential forms 729.1 p-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729.2 Integration on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749.3 Submanifolds, Stokes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

10 The initial value problem 7610.1 Extrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7610.2 The Gauss-Codazzi equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.3 The constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.4 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7910.5 The 3+1 equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

11 The Lagrangian formulation 82

1 THE EQUIVALENCE PRINCIPLE 4

1 The equivalence principle

Special Relativity: • Physical experiments are the same in any inertial frame

• inertial frames: non-accelerating observers

related by Lorentz trafos

Newtonian gravity: ∇2φ = 4πGρ (∗)

⇒ φ(t, ~x) = −G∫

ρ(t, ~y)

|~x− ~y| d3y

Lorentz trafos mix time and space coordinates

also: finite propagation of signals

⇒ Eq. (∗) not invariant

⇒ Newton’s gravity not compatible with SR

Newtonian gravity: good approximation if v ≪ c

orbiting particle: φ = −GMr

⇒ v2

r=GM

r2

then: v ≪ c ⇔ G

c2M

r≪ 1

Solar system:G

c2M

r< 10−5

m

M

r

1.1 Statement of the equivalence principle

Newtonian theory

Inertial mass: ~F = mI~a

Gravitational mass: ~F = −mG~∇φ = mG~g ; ~g := −~∇φ

⇒ With suitable scaling: mI = mG

Experiment:mI

mG

− 1 = O(10−12) “Eotvos” for all kinds of objects

Weak equivalence principle (WEP), version 1: mI = mG

Newtonian motion: mI ~a = mI ~x = mG ~g ⇒ ~x = ~g


⇒WEP, version 2: The trajectory of a freely falling test body

depends only on its initial position and velocity

and is independent of its composition.

Comment: “Test body” = body with negligible gravitational self interaction

and size ≪ lengthscale on which ~g varies.

accelerated frames

Let O be inertial frame with coords. (t, ~x) in grav. field ~g .

Let O′ be a frame accelerated relative to O with ~a .

Coords.:(t, ~x′ = ~x− ~x0(t)

)where ~x0(t) = position of origin of O′ in O coords.: ~x0 = ~a

⇒ Eq. of motion in O′ : ~x′ = ~g − ~a

→ different grav. field ~g′ = ~g − ~a

special cases: 1) ~g = 0 ⇒ ~g′ = −~a

“uniform acceleration indistinguishable from grav. field”

2) ~g 6= 0 , ~a = ~g ⇒ O′ is a freely falling frame: ~g′ = 0

Non-uniform grav. fields

“local inertial frame” = coord. frame (t, x, y, z) defined by freely falling observer in

the same way as in Minkowski space

“local” means “small compared with lengthscale of variations in ~g

E.g.: tidal forces:

Lab frame “too large”

⇒ particles accelerated in freely

falling frame due to tidal forces. Earth

Lab frame


The WEP was found in Newtonian physics

Einstein promoted it to be more general:

Einstein EP (EEP): (i) The WEP is valid.

(ii) In a local inertial frame the results of all non-gravitational

experiments are indistinguishable from those of the same

experiment performed in an inertial frame in Minkowski spacetime.

Schiff’s conjecture: The WEP implies the EEP.

argument: • WEP ⇒ (ii) holds for test particles.

• Matter is composed of quarks, electrons etc.

• These are bound by electromagnetic, nuclear forces

→ binding energy makes up part of the

bodies mass and appears to also obey (ii)

1.2 Bending of light

Consider freely falling lab in uniform grav. field

Earth

Lab

Inside Lab

light moves on straight line

Light

Earth frame

curved path

Earth

d=ct

gt /22

t =d

c⇒ h =

1

2gd2

c2

d = 1 km ⇒ h ≈ 5 · 10−11 m


1.3 Gravitational redshift

Consider: ~g = (0, 0, −g) , Alice at z = h, Bob at z = 0

Alice sends light to Bob.

Bobx, y

gz

Alice

EP ⇒ equivalent to frame accelerated with (0, 0, +g) in

Minkowski spacetime

Assumption: v of Bob, Alice ≪ c

⇒ ignorev2

c2and higher-order SR terms

⇒ zA(t) = h+1

2gt2 , zB(t) =

1

2gt2 , vA = vB = gt

!≪ c .

• Alice emits first signal at t1

⇒ z1(t) = zA(t1)− c(t− t1) = h+1

2gt21 − c(t− t1)

• This reaches Bob at T1, i.e. h+1

2gt21 − c(T1 − t1) =

1

2gT 2

1 (∗∗)

• Alice emits second signal at t2 = t1 +∆τA .

This reaches Bob at T2 = T1 +∆τB .

⇒ h +1

2g(t1 +∆τA)

2 − c(T1 +∆τB − t1 −∆τA) =1

2g(T1 +∆τB)

2

∣∣∣∣

subtract (∗∗)

⇒ c(∆τA −∆τB) +1

2g∆τA (2t1 +∆τA) =

1

2g∆τB (2T1 +∆τB)

• Assumption: ∆τA ≪ t1 , ∆τB ≪ T1 , e.g. period in light waves

⇒ c(∆τA −∆τB) + g∆τA t1 = g∆τB T1

⇒ ∆τB (gT1 + c) = ∆τA (gt1 + c)

⇒ ∆τB =

(

1 +gT1c

)−1(

1 +gt1c

)

∆τA ≈[

1− g(T1 − t1)c

]

∆τA

∣∣∣∣

we usedgt

c≪ 1

• (∗∗) ⇒ h

c− (T1 − t1) =

1

2

g

c(T1 + t1)︸︷︷︸

≪ 1

(T1 − t1)︸︷︷︸

≈ h

c

≈ 0

∣∣∣∣

we usedgt

c≪ 1

⇒ T1 − t1 =h

cto leading order.


• ⇒ ∆τB ≈(

1− gh

c2

)

∆τA!< ∆τA

⇒ Signal appears blue shifted to Bob: c∆τB = λB ≈(

1− gh

c2

)

λA

Confirmed in Pound-Rebka experiment (1960): light falling in tower.

Light climbing out of a gravity well is red shifted.

In general: ∆τB ≈(

1 +φB − φA

c2

)

∆τA also holds for weak, non-uniform fields

1.4 Curved spacetime

WEP ⇒ test bodies move the same way in a grav. field independent of their

composition, i.e. their grav. “charge” m. This is not true for other forces!

Einstein: gravity must be a feature of spacetime, i.e. its geometry.

Consider redshift but now in a non-Minkowskian metric

c2 dτ 2 =

[

1 +2φ(x, y, z)

c2

]

c2dt2 −[

1− 2φ(x, y, z)

c2

]

(dx2 + dy2 + dz2) ;φ

c2≪ 1

• Alice: ~xA , Bob: ~xB , at fixed positions!

• Alice emits signals at tA, tA +∆t

Bob receives the first at tB . When does he see the second?

• The spacetime is static: φ does not depend on t

⇒ The two signals travel on identical trajectories, just shifted in time

⇒ Bob receives the second signal at tB +∆t .

• But what proper times do Alice’s and Bob’s clocks measure?

∆τ 2A =

(

1 +2φAc2

)

∆t2 ,

⇒ ∆τA ≈(

1 +φAc2

)

∆t ,

∆τ 2B =

(

1 +2φBc2

)

∆t2

⇒ ∆τB ≈(

1 +φBc2

)

∆t ,

⇒ ∆τB ≈(

1 +φBc2

) (

1 +φAc2

)−1

∆τA ≈(

1 +φB − φA

c2

)

∆τA

2 MANIFOLDS AND TENSORS 9

2 Manifolds and tensors

In GR we define spacetime as a manifold: trickier than for Minkowski!

Minkowski: • inertial frames → preferred global coordinates

• we can add position vectors ⇒ spacetime has structure of vector space

Curved spacetimes: inertial coordinates are local; how about vectors?

2.1 Differentiable manifolds

We know how to do calculous in Rn

Goal: develop analog in curved spaces

Def.: n-dim. differentiable manifold := a setM with subsets Oα such that

(1) ∪αOα =M

(2) ∀α ∃ a 1-to-1 and onto map

φα : Oα −→ Uα ⊂ Rn open

(3) If Oα ∩Oβ 6= ∅ , then φβ φ−1α : [φα(Oα ∩ Oβ)] −→ [φβ (Oα ∩ Oβ)]

տ ⊂ Uα ⊂ Rn տ ⊂ Uβ ⊂ R

n

is a smooth map (∞ differentiable)

The φα are called “charts”

φα is an “atlas”

Oα

Oβ

M

Uα

Uβ

φα(Oα ∩ Oβ) φβ(Oα ∩ Oβ)

φαφβ

φβ φ−1α


Comments: • For p ∈ Oα we often write φα(p) =(

x1α(p), x2α(p), x

3α(p)

)

= xµα(p)

= “coordinates” of p ; the α is often dropped.

• A Ck manifold is defined likewise. We’ll assume C∞

Examples: 1) Rn is a manifold with an atlas of one chart

φ : (x1, . . . , xn) 7→ (x1, . . . , xn)

2) S1 ≡ unit circle =(cos θ, sin θ) ∈ R

2∣∣ θ ∈ R

∃ no atlas with one chart

θ ∈ [0, 2π) does not work: not open!

We need 2 charts:

(i) Let P = (1, 0) and φ1 : S1 − P → (0, 2π), φ1(p) = θ1

x

yp

Q θ2

y

x

p

Pθ1

(ii) Let Q = (−1, 0) and φ2 : S1 − Q → (−π, π), φ2(p) = θ2

φ1, φ2 form an atlas

Note: On the upper semi circle (y ≥ 0): θ2 = φ2 φ−11 (θ1) = θ1

On the lower semi circle (y < 0): θ2 = φ2 φ−11 (θ1) = θ1 − 2π

Comment: M may admit many atlases

Def.: 2 atlases are compatible :⇔ their union is also an atlas

complete atlas := union of all atlases compatible with a given atlasտ contains ∞ atlases


2.2 Smooth functions

Def.: f :M→ R is smooth :⇔ ∀ charts φ : F ≡ f φ−1 : U ⊂ Rn → R is smooth

Sometimes we call f a scalar field

Examples: 1) Consider the S1 sphere above: f : S1 → R, (x, y) 7→ x

f φ−11 (θ1) = cos θ1 , f φ−1

2 (θ2) = cos θ2 both smooth

Let φ be some chart ⇒ f φ−1 =(f φ−1

i

)

︸︷︷︸

smooth

(φi φ−1

)

︸︷︷︸

smooth

(manifold!)

, i = 1, 2

2) Consider manifoldM, chart φ : O ⊂M→ U ⊂ Rn , p ∈ O 7→

(x1(p), . . . , xn(p)

)

Let φα be the other charts in the atlas

Let f : O → R , p 7→ x1(p)

⇒ f is smooth: x1 φ−1α is the first component of φ φ−1

α which is smooth

3)We can define f through F :

φα atlas ⇒ Fα : Uα → R defines f = Fα φαprovided Fα is independent of α on overlaps

Consider S1 above: F1 : (0, 2π)→ R , θ1 7→ sin(mθ1) , m integer

F2 : (−π, π)→ R , θ2 7→ sin(mθ2)

⇒ F1 φ1 = F2 φ2 on overlap: θ1, θ2 differ by multiples of 2π

Note: We sometimes do not distinguish between f and F : “f(x) = F (x)”


2.3 Curves and vectors

Consider surface S in R3, tangent plane at p

⇒ the plane has structure of a 2-dim. vector space;

a tangent vector to a curve in S at p is in the plane

p

Goal: formalize this for a manifold

Def.: A smooth curve in a manifoldM := function λ : I →M , where I ⊂ R open,

such that φα λ : I → Rn is smooth for all charts φα

Directional derivative: let f :M→ R , λ : I →M both be smooth

⇒ f λ : I → R smooth

⇒ d

dt[(f λ) (t)] = d

dt[f(λ(t))]

Def.: Let C∞ be the space of all smooth functions fromM to R.

Let λ be a smooth curve with λ(0) = p ∈M

⇒ The “tangent vector” to λ is the linear map

Xp : C∞(M,R)→ R , f 7→Xp(f) =

d

dt[f(λ(t))]

t=0

Note: (i) Linearity ⇒ Xp(f+g) = Xp(f)+Xp(g) ; Xp(αf) = αXp(f) for α = const

(ii) Xp(f g) = Xp(f) g(p) + f(p)Xp(g) “Leibniz rule”

(iii) Let φ = (xµ) be a chart defined in a neighbourhood of p ∈M, and F ≡ f φ−1

⇒ f λ = (f φ−1) (φ λ) = F φ λ ,

Xp(f) =

(∂F

∂xµ

)

φ(p)

(d xµ(λ(t))

dt

)

t=0

=dxµ

dt︸︷︷︸

∂

∂xµ︸︷︷︸

F =d

dt︸︷︷︸

F (xµ(λ(t))) =d

dtf(λ(t))

ր ↑ տcomponents basis vector

Compare with directional derivative in Rn: ~X · (~∇f)p


The set of tangent vectors at p ∈M forms an n-dim. vector space: “Tangent Space” Tp(M)

Proof: (1) “Addition, scalar mult. → vector”

Let λ, κ be curves through p such that λ(0) = κ(0) = p,

Xp, Y p be their tangent vectors,

α, β ∈ R, φ = (xµ) be a chart in neighbourhood of p.

Define αXp + βY p : C∞(M,R)→ R , f 7→ αXp(f) + βY p(f)

Consider curve ν(t) ≡ φ−1α[φ(λ(t)

)− φ(p)

]+ β

[φ(κ(t)

)− φ(p)

]+ φ(p)

⇒ ν(0) = p

Let Zp be the tangent vector of ν

⇒ Zp(f) =

(∂F

∂xµ

)

φ(p)

d

dt

[α(xµ(λ(t))− xµ(p)

)+ β

(xµ(κ(t))− xµ(p)

)]

t=0

=

(∂F

∂xµ

)

φ(p)

α

[dxµ(λ(t))

dt

]

t=0

+ β

[dxµ(κ(t))

dt

]

t=0

= αXp(f) + βY p(f) = (αXp + βY p)(f)

(2) “n dims.?”

Define for µ = 1, . . . , n:

λµ(t) ≡ φ−1[x1(p), . . . , xµ−1(p), xµ(p) + t, xµ+1(p), . . . , xn(p)

]

Let

(∂

∂xµ

)

p

be the tangent vector to λµ

⇒(

∂

∂xµ

)

p

(f) =∂F

∂xµ

∣∣∣∣φ(p)

(∗)

Let αµ ∈ R such that αµ(

∂

∂xµ

)

p

= 0 ∈ Tp(M)

⇒ αµ(∂F

∂xµ

)

φ(p)

= 0

Let F (xµ) = xν ⇒ ∂F

∂xµ= δνµ ⇒ αν = 0 .

Do this for all ν = 1, . . . , n ⇒ lin. independence.


(3) “Do we span Tp(M)?”

Xp(f) =

(dxµ(λ(t))

dt

)

t=0

(∂

∂xµ

)

p

(f) for any f !

⇒ Xp =

(dxµ(λ(t))

dt

)

t=0

(∂

∂xµ

)

p

(∗∗)

⇒ Any Xp can be written as a linear combination of

(∂

∂xµ

)

p

.

Note: •(

∂

∂xµ

)

p

: C∞(M,R)→ R is not the same as the partial derivative∂

∂xµ!

• The basis

(∂

∂xµ

)

p

is chart dependent: “coordinate basis”

Def.: Let eµ , µ = 1, . . . , n be a basis of Tp(M)

⇒ Xp = Xµp eµ ; Xµ

p are the “components” of Xp

Example: (∗∗) for coord. basis: Xµp =

[dxµ (λ(t))

dt

]

t=0

=:dxµ

dt

Note: When Einstein summation applies: always one index up, one down !(

∂

∂xµ

)

p

is a “down” index. Expressions like XµYµ are wrong !

Coordinate transformations

Let φ = (xi) , φ = (xi) be two charts in a nbhd. of p ∈M


⇒(

∂

∂xµ

)

p

(f) =∂

∂xµ(f φ−1)

∣∣∣∣φ(p)

=∂

∂xµ

[ (f φ−1

)(φ φ−1

︸︷︷︸

= xµ(xα)

) ]∣∣∣∣φ(p)

=∂

∂xµ

[(

f φ−1)(x(x)

)]

=

[∂

∂xα(f φ−1

)(x)

]

φ(p)

∂xα

∂xµ

∣∣∣∣φ(p)

=

(∂

∂xα

)

p

(f)∂xα

∂xµ

∣∣∣∣φ(p)

⇒(

∂

∂xµ

)

p

=

(∂xα

∂xµ

)

φ(p)

(∂

∂xα

)

p

Components: V ∈ Tp(M)

⇒ V = V µ

(∂

∂xµ

)

p

= V µ

(∂xα

∂xµ

)

φ(p)

(∂

∂xα

)

p

= V α

(∂

∂xα

)

p

⇒ V α =

(∂xα

∂xµ

)

V µ

V µ = components of V in basis

(∂

∂xµ

)

V α = components of V in basis

(∂

∂xα

)


2.4 Covectors

Def.: Let V be a vector space over R.

“Dual space” V∗ := vector space of linear maps from V to R

Lemma: V n-dimensional ⇒ V∗ n-dim.

if eµ , µ = 1, . . . , n is a basis of V

⇒ fα , α = 1, . . . , n , defined by fα(eµ) = δαµ , is the “dual basis” of V∗

Comments: • V, V∗ are isomorphic ; e.g. eµ 7→ fµ defines an isomorphism

• The isomorphism is basis dependent

• There is a natural isomorphism between V and (V∗)∗

Theorem: If V is finite dim.

⇒ A natural, basis independent isomorphism is given by

Φ : V → (V∗)∗ , X 7→ Φ(X) with(Φ(X)

)(ω) := ω(X) ∀ω∈V∗

Def.: “Cotangent space” T ∗p (M) := dual space of Tp(M)

Its elements are “covectors” or “1-forms”

If eµ is a basis of Tp(M) and fµ the dual basis in T ∗

p (M)

⇒ η = ηµfµ ∈ T ∗

p (M) ; ηµ are the “components” of η

Comments: • η(eµ) = ηνfν(eµ) = ηµ

• X ∈ Tp(M) ⇒ η(X) = η(Xµeµ) = Xµη(eµ) = Xµηµ

Def.: Let f :M→ R be a smooth function

“gradient of f” at p := (df)p ∈ T ∗p (M) with

(df)p(X) := X(f) ∀ X ∈ Tp(M)


Examples (1) Let (xi) be a coord. chart in nbhd. of p ∈M and f := xµ(p) for some µ

⇒(dxµ

)

p∈ T ∗

p (M) with(dxµ

)

p

((∂

∂xν

)

p

)

=∂xµ

∂xν

∣∣∣∣p

= δµν

⇒(

dxµ)

p

is the dual basis of

(∂

∂xµ

)

p

(2) Components of(df)

p:

[(df)

p

]

µ=(df)

p

((∂

∂xµ

)

p

)

=

(∂

∂xµ

)

p

(f) =

(∂F

∂xµ

)

φ(p)

Coordinate transformation

If φ = (xi) , φ = (xi) are two charts in nbhd. of p ∈M

⇒ . . . ⇒(dxµ

)

p=

(∂xµ

∂xν

)

φ(p)

(dxν)

p

so for ω ∈ T ∗p (M): ω = ωµ dx

µ = ωµ dxµ with ωµ =

(∂xν

∂xµ

)

φ(p)

ων “covariant vector”

2.5 Abstract index notation

We have used µ, ν, . . . for components of vectors or 1-forms in a basis

Some expressions are basis dependent, some are not!

E.g.: η(X) = ηµXµ independent

Xµ = δµ1 dependent

Index notation: If a statement is true in any basis, replace µ ν, . . . with a, b, . . .

E.g.: η(X) = ηaXa

Convention: a, b, . . . do not denote components, but place holders for component indices

Xa is a vector, ηa a 1-form, . . . ; “Xa 6= Xµ”

The rules for index positions are the same as for µ, ν, . . . . E.g. ηaωa is wrong


2.6 Tensors

Tensors in physics: e.g. moment of inertia

In GR many things are tensors

Def.: A tensor of type (r, s) or(rs

)is a multilinear map

T : T ∗p (M)× . . .× T ∗

p (M)︸︷︷︸

r factors

×Tp(M)× . . .× Tp(M)︸︷︷︸

s factors

→ R

A machine: input: r 1-forms, s vectors ; output: a real number

Examples: (1) 1-form = (0, 1) tensor : Tp(M)→ R

(2) Recall:(T ∗p (M)

)∗is naturally isomorphic to Tp(M)

⇒ vector = (1, 0) tensor : T ∗p (M)→ R , η 7→ η(X) ∀ η ∈ T ∗

p (M)

(3) Define the (1, 1) tensor δ : T ∗p (M)× Tp(M)→ R through

δ(η,X) := η(X) ∀ η ∈ T ∗p (M) , X ∈ Tp(M)

Def.: Let eµ be a basis of Tp(M) and fν the dual basis of T ∗p (M)

The “components of a (r, s) tensor” T are

T µ1µ2...µrν1ν2...νs := T (fµ1 , fµ2 , . . . , fµr , eν1, eν2 , . . . , eνs)

In abstract index notation: T a1a2...ar b1b2...bs

Comment: Tensors of type (r, s) in p ∈M can be added or multiplied by constants.

They form a vector space of dimension nr+s

Examples: (1) δ above: δµν = δ(fµ, eν) = fµ(eν) = δµν

(2) Let η , ω ∈ T ∗p (M) , X ∈ Tp(M) , T a (2, 1) tensor, eµ, fν bases

⇒ T (η, ω, X) = T (ηµfµ, ωνf

ν , Xαeα)

= ηµωνXαT (fµ, fν , eα) = ηµωνX

αT µνα

Index notation: ηaωbXcT abc


Change of basis

Let eµ, eν be bases of Tp(M) and fµ, fν the dual bases of T ∗p (M)

Transformation matrices: fµ= Aµνf

ν , eµ = Bνµeν

We have: δµν = fµ(eν) = Aµρf

ρ(Bσνeσ) = AµρB

σν f

ρ(eσ)︸︷︷︸

= δρσ

= AµρBρν

⇒ Bµν =

(A−1

)µν are inverses !

E.g. coord. basis: Aµν =∂xµ

∂xν, Bν

µ =∂xν

∂xµare obviously inverses

One straightforwardly shows: vector: Xµ = AµνXν

1-form: ηµ = (A−1)νµην

(2, 1) tensor: T µνρ = AµαAνβ (A

−1)γρ T

αβγ

(r, s) tensor: obvious...

Def.: “Contraction of (r, s) tensor” := Summation over 1 upper and 1 lower index

→ (r − 1, s− 1) tensor

Example: Let T be a (3, 2) tensor

⇒ (2, 1) tensor S(ω,η,X) := T (fµ,ω,η, eµ,X)

This is basis independent:

T (fµ,ω,η, eµ,X) = T

(Aµνf

ν ,ω,η,(A−1

)ρµeρ,X

)= T (fν ,ω,η, eν ,X)

Components: Sµνρ = T αµναρ

Abstract index notation: Sabc = T dabdc

Note: In general T dabdc 6= T abddc ⇒ index position important !


Def.: Let S be a (p, q) tensor, T a (r, s) tensor

“outer product” S ⊗ T is a (p+ r, q + s) tensor with(S ⊗ T

)(ω1, . . . , ωp, η1, . . . , ηr, X1, . . . , Xq, Y 1, . . . , Y s)

:= S(ω1, . . . , ωp, X1, . . . , Xq)T (η1, . . . , ηr, Y 1, . . . , Y s)

One straightforwardly shows:

(1)(S ⊗ T

)a1...apb1...brc1...cqd1...ds = Sa1...apc1...cq T

b1...brd1...ds

(2) In a coord. basis, a (2, 1) tensor can be written as

T = T µνρ

(∂

∂xµ

)

p

⊗(

∂

∂xν

)

p

⊗(dxρ)

p

likewise (r, s) tensor

Comment: We always first put in 1-forms into a tensor, then vectors.

This is not necessary. We can define

T : T ∗p × Tp × T ∗

p → R , (η,X,ω) 7→ T (η,X,ω)

and this is isomorphic to

T : T ∗p × T ∗

p × Tp → R , (η,ω,X) 7→ T (η,ω,X) .

So we do not distinguish between them.

But be careful with index positions: In general T abcηaωb = T bacηbωa 6= T bacηaωb


Def.: Let T be a (0, 2) tensor.

“Symmetrization”: Sab :=1

2(Tab + Tba) =: T(ab)

“Anti-symmetrization”: Aab :=1

2(Tab − Tba) =: T[ab]

Can be applied to a subset of indices: T (ab)cd =

12(T abcd + T bacd)

Over n > 2 indices: • sum over all permutations

• apply sign of permutation for anti-symm.

• divide by n!

E.g.: T a[bcd] =1

3!

(T abcd + T adbc + T acdb − T adcb − T acbd − T abdc

)

For non-adjacent indices: T(a|bc|d) :=1

2

(Tabcd + Tdbca

)

2.7 Tensor Fields

So far: tensors at point p ∈M ; Now: fields

Def.: vector field := a map X :M→ Tp(M) , p→Xp

Let f :M→ R be smooth

⇒ X(f) is a function X(f) :M→ R , p 7→Xp(f)

X is smooth :⇔ X(f) smooth for all smooth f

Example: Let φ = (xµ) be a chart and ∂µ :=

(∂

∂xµ

)

be the vector field defined by p 7→(

∂

∂xµ

)

p

⇒ ∂µ(f) :M→ R , p 7→(∂F

∂xµ

)

φ(p)

where F = f φ−1

Note: Everything is smooth: φ(p), F (xµ),∂F

∂xµ⇒ ∂µ(f) :M→ R is smooth.

φ may only cover a subset of M and the map only part of M→ φα on patches Oα


Comment:

(∂

∂xµ

)

p

is basis of Tp

⇒ Expand vector field: X = Xµ

(∂

∂xµ

)

= Xµ∂µ

∂µ smooth ⇒(X smooth ⇔ Xµ smooth functions

)

Def.: Covector field ω := map :M→ T ∗p (M) , p 7→ ωp

Note: a vector field X and covector field ω define a function

ω(X) :M→ R , p 7→ ωp(Xp)

ω smooth :⇔ ω(X) is smooth for all smooth X

Example: df :M→ T ∗p (M) , p 7→ (df)p

f, X smooth ⇒ df(X) = X(f) is a smooth function

⇒ df is smooth, “gradient”

Set f = xµ ⇒ dxµ is a smooth covector field

Def.: (r, s) Tensor field := map T :M→ (r, s) tensor at p ∈M

Smooth vector, covector fields η1, . . . , ηr, X1, . . . , Xs define a function

T (η1, . . . , ηr, X1, . . . , Xs) :M→ R , p 7→ T p

(

(η1)p, . . . , (ηr)p, (X1)p, . . . , (Xs)p

)

T smooth :⇔ this function is smooth ∀ smooth η1, . . . , ηr, X1, . . . , Xs

Note: One can show: T smooth ⇔ its components in coord. basis are smooth

From now on: assume all our tensors are smooth


2.8 The commutator

Let X, Y be vector fields, f, g functions

⇒ Y (f) is a function ⇒ X(Y (f)

)is a function

But: X(Y (f g)

)= X

(f Y (g) + g Y (f)

)

= fX(Y (g)

)+X(f)Y (g) +X(g)Y (f) + gX

(Y (f)

)

6= fX(Y (g)

)+ gX

(Y (f)

)“no Leibniz”!

⇒ The map f 7→ X(Y (f)

)does not define a vector field. But: ...

Def.: Commutator of 2 vectorfields X, Y :

[X,Y ](f) := X(Y (f)

)− Y

(X(f)

)satisfies Leibniz!

[X,Y ] is indeed a vectorfield

Proof: coord. chart (xµ)

⇒ [X ,Y ](f) = X

(

Y ν ∂F

∂xν

)

− Y

(

Xµ ∂F

∂xµ

)

= Xµ ∂

∂xµ

(

Y ν ∂F

∂xν

)

− Y ν ∂

∂xν

(

Xµ ∂F

∂xµ

)

= Xµ∂Yν

∂xµ∂F

∂xν− Y ν ∂X

µ

∂xν∂F

∂xµ

=

(

Xν ∂Yµ

∂xν− Y ν ∂X

µ

∂xν

)

︸︷︷︸

[X, Y ]µ :=

∂F

∂xµ

f arbitrary ⇒ [X ,Y ] = [X, Y ]µ(

∂

∂xµ

)

Example: Let X =∂

∂x1, Y = x1

∂

∂x2+

∂

∂x3

⇒ [X, Y ]µ =∂Y µ

∂x1= δµ2

⇒ [X ,Y ] =∂

∂x2


One can show: [X,Y ] = −[Y ,X]

[X,Y +Z] = [X,Y ] + [X +Z]

[X, f Y ] = f [X,Y ] +X(f)Y

[X, [Y ,Z]] + [Y , [Z,X]] + [Z, [X,Y ]] = 0 “Jacobi identity”

Note:

[∂

∂xµ,∂

∂xν

]

= 0 (coord. basis ⇒ commutators vanish)

Conversely, one can show:

If X1, . . . , Xm , m ≤ dim(M) are vector fields which are

lin. indep. ∀p ∈ M and whose commutators all vanish

⇒ In a nbhd. of p one can find coords. (xµ)

such that X i =∂

∂xi, i = 1, . . . , m

2.9 Integral curves

Def.: Let X be a VF and p ∈M.

“integral curve of X through p”

:= curve through p whose tangent at every point q is Xq

Let λ be an integral curve of X , λ(0) = p , (xµ) be a coord. chart

⇒ dxµ(λ(t)

)

dt= Xµ

(

xα(λ(t)

))

, xµ(λ(0)

)= xµp (∗)

ODE theory guarantees existence, uniqueness of solution

⇒ ∃ unique integral curve of X through p ∈M

3 THE METRIC TENSOR 25

Example: Chart φ = (xµ) , let X =∂

∂x1+ x1

∂

∂x2, xµ(p) = (0, . . . , 0)

(∗) ⇒ dx1

dt= 1 ,

dx2

dt= x1

⇒ . . .⇒ x1 = t , x2 =t2

2, xi = 0 for i = 3, . . . , n

3 The metric tensor

3.1 Metrics

We want to measure things → need metric!

E.g.: R3, scalar product: maps 2 vectors to R

⇒ metric should be (0, 2) tensor

Def.: A metric at p ∈M := (0, 2) tensor that is:

(i) symmetric: g(X,Y ) = g(Y ,X) ∀ X, Y ∈ Tp(M) ⇔ gab = gba

(ii) non-degenerate: g(X,Y ) = 0 ∀ Y ∈ Tp(M) ⇔ X = 0

Notation: g(X,Y ) = 〈X,Y 〉 = X · Y

Comment: a metric defines an isomorphism between vectors and 1-forms:

X 7→ g(X, .) =: X , i.e. X : Tp(M)→ R , Y 7→X(Y ) := g(X,Y )

with the metric inverse (see below), we can raise and lower indices of tensors


Signature

g symmetric ⇒ components of g at p ∈M are a symmetric matrix

⇒ ∃ basis where gµν is diagonal

g non-degenerate ⇒ all diagonal elements are 6= 0

⇒ we can rescale the basis such that the diagonal elements = ±1

“orthonormal basis” ← basis non-unique!

“Sylvester’s law” ⇒ the number of +1 and −1 entries is independent of basis

Def.: “signature” := sum +1, −1 over all diagonal elements

Riemannian metrics: signature = + + . . .+ or +n = # of dims.

Lorentzian metrics: −++ . . .+ or n− 2. Some people use +−− . . .−

Note: Equivalence principle

⇒ in a local inertial frame, the laws of SR hold

⇒ ∃ chart: metric gµν = ηµν = diag(−1, 1, 1, 1) “Lorentz invariant”

Only possible locally! At q 6= p, gµν 6= ηµν in general

Def.: “A Riemannian (Lorentzian) manifold”

:= (M, g) where M is a diff. manifold and g a Riemannian (Lorentzian) metric

“spacetime” := Lorentzian manifold

Notation: in coord. basis: g = gµν dxµ ⊗ dxν

often used: ds2 = gµνdxµdxν


Comment: Let λ : (a, b) ⊂ R→M be a smooth curve on a Riemannian manifold,

X be the tangent vector of λ

Then the length of λ is:

∫ b

a

√

g(X,X)λ(t)dt

re-parametrize: t = t(u) withdt

du> 0 , u ∈ (c, d) , t(c) = a , t(d) = b

⇒ the curve κ(u) := λ(t(u)

)has tangent vector Y =

dt

duX

⇒ the length of κ is the same as that of λ.

Examples

(1) Euclidean metric in Rn with coords. x1, . . . , xn:

g = dx1 ⊗ dx1 + . . .+ dxn ⊗ dxn .

A coord. chart of (Rn, g) where gµν = diag(1, . . . , 1) is called “Cartesian”

(2) Minkowski metric in R4 with coords. x0, x1, x2, x3:

η = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3)2 , (dx0)2 ≡ dx0 ⊗ dx0 , . . .

A coord. chart which covers R4 such that ηµν = diag(−1, 1, 1, 1) is

called “inertial frame”. (R4,η) =: Minkowski spacetime

(3) Let (θ, φ) be spherical coords. on S2 ⇒ ds2 = dθ2 + sin2 θ dφ2

This is positive definite on θ ∈ (0, π) but not on all S2

⇒ We need second chart, e.g. θ′, φ′ with

x = − sin θ′ cosφ′ , y = cos θ′ , z = sin θ′ sinφ′

Def.: g non-degenerate ⇒ g invertible

“inverse metric” = g−1 := symmetric (2, 0) tensor gab with gabgbc = δac

Example: On S2 on the chart (θ, φ) we have gµν = diag(1, 1/ sin2 θ)


Comment: g−1 maps 1-forms to vectors:(g−1(η, . )

)(ω) := g−1(η,ω)

The metric mappings between vectors and 1-forms are inverses of each other:

g−1(g(X, . ), .

)= X , g

(g−1(η, . ), .

)= η

→ natural isomorphism

Example: Let T be a (3, 2) tensor: T abcde = gbfg

dhgejT afchj

− we use the same letter T irrespective of the up or down position of indices

− the order of indices is preserved!

3.2 Lorentzian signature

Note: indices typically chosen to run from 0 . . . 3

At any p ∈M of a Lorentzian manifold:

we can choose orthonormal basis (ONB): g(eµ, eν) = ηµν = diag(−1, 1, 1, 1)

This basis is not unique:

eµ =(A−1

)νµeν

⇒ gµν = g(eµ, eν) =(A−1

)ρµ

(A−1

)σν g(eρ, eσ) =

(A−1

)ρµ

(A−1

)σν ηρσ

!= ηµν

⇒ AµρAνσηµν = ηρσ “Lorentz trafos of SR!

⇒ ONBs are related by Lorentz trafos

⇒ locally at p we recover SR

Def.: Let (M, g) be a Lorentzian manifold, X ∈ Tp(M), X 6= 0

X is timelike :⇔ g(X,X) < 0

null :⇔ g(X,X) = 0

spacelike :⇔ g(X,X) > 0

In an ONB, gµν = ηµν locally

⇒ locally we have the light cone structure of SR

null

spacelike

timelike


One can show: If X, Y ∈ Tp(M), X, Y 6= 0 with g(X,Y ) = 0. Then

X timelike ⇒ Y spacelike

X null ⇒ Y spacelike or null

X spacelike ⇒ Y spacelike, timelike or null

Principle of proof: Apply spatial rotation such that X has simple space components.

E.g. timelike X → Xµ = (X0, X1, 0, 0)

Def.: On a Riemannian manifold:

“norm” of X ∈ Tp(M): |X| :=√

g(X,X)

“angle” between X, Y ∈ Tp(M): θ := arccos

(g(X,Y )

|X| |Y |

)

Same for spacelike vectors in Lorentzian manifold.

Def.: A curve is timelike (null, spacelike)

:⇔ its tangent vector is timelike (null, spacelike) everywhere

Comments: • curves often change their character between timelike, null, spacelike

• the length of a spacelike curve λ is

s =

∫ t1

t0

√

g(X,X)|λ(t)dt

• for timelike curves we define the proper time along the curve as

τ :=

∫ t1

t0

√

− g(X,X)|λ(t)dt

In a coord. chart: Xµ =dxµ

dt, so we often write:

dτ 2 = −gµνdxµdxν , τ =

∫

dτ


Def.: “4-velocity” of a timelike curve λ

:= tangent vector of the curve parametrized by the proper time

uµ =dxµ

dτ

∣∣∣∣λ(τ)

Note: along this curve:

τ =

∫ τ1

τ0

√

−gµνdxµ

dτ

dxν

dτdτ =

∫ τ1

τ0

√−gµνuµuνdτ

∣∣∣∣∣

d

dτ

⇒ 1 =√−gµνuµuν

⇒ gµνuµuν = −1

3.3 Curves of extremal proper time

Let p, q ∈M be connected by a timelike curve λ

A small deformation of λ is still timelike

Which curve connecting p, q extremizes the proper time along it?

Let u be a parameter such that λ(u = 0) = p, λ(u = 1) = q . Let ˙ :=d

du.

⇒ τ [λ] =

∫ 1

0

G(x(u), x(u)

)du with G =

√

−gµν(x(u)) xµ(u) xν(u)

and x(u) := x(λ(u)

)

This is an Euler-Lagrange problem

⇒ the extremal curve satisfiesd

du

(∂G

∂xµ

)

− ∂G

∂xµ= 0

We have:∂G

∂xµ= − 1

2G2gµν x

ν = − 1

Ggµν x

ν

∂G

∂xµ= − 1

2G∂µgνρ x

ν xρ , where ∂µ :=∂

∂xµ

Now change to the proper time as a parameter: τ =

∫ √

−gµνdxµ

du

dxν

dudu


⇒(dτ

du

)2

= −gµν xµxν = G2

⇒ dτ

du= G

⇒ d

du= G

d

dτ

⇒ . . .⇒ Euler-Lagrange Eq.:d

dτ

(

gµνdxν

dτ

)

− 1

2∂µgνρ

dxν

dτ

dxρ

dτ= 0

⇒ gµνd2xν

dτ 2+ ∂ρgµν

︸︷︷︸

!= ∂ρg(µν)

dxρ

dτ

dxν

dτ︸︷︷︸

symm. in ρ, ν

−12∂µgνρ

dxν

dτ

dxρ

dτ= 0

∣∣∣∣∣· gαµ

⇒ d2xα

dτ 2+ Γανρ

dxν

dτ

dxρ

dτ= 0 (∗)

with Γανρ =1

2gαµ (∂ρgµν + ∂νgρµ − ∂µgνρ) “Christoffel symbols”

Comments: • Γανρ = Γαρν

• Γανρ are not tensor components

• The individual terms of (∗) are not vector components, but the sum is

• (∗) is called the “geodesic equation”

• In Minkowski: Γανρ = 0 ⇒ d2xα

dτ 2= 0

⇒ The eqs. of motion of a free particle extremize proper time

Postulate: Massive particles in GR follow curves of extremal proper time,

i.e. follow (∗)

Comments: • massless particles follow a similar equation

• In Minkowski: curves of extremal proper time maximize proper time

between 2 points.

In GR: This holds locally; the max. may not be a global one.


One can show the following:

(1) (∗) are the Euler-Lagrange eqs. of L = −gµν(x(τ)

) dxµ

dτ

dxν

dτ

→ easy way to calculate Γανρ

(2) L has no explicit τ dependence:∂L

∂τ= 0

with EL eqs.: ⇒ . . .⇒ L− ∂L

∂xµxµ = gµν x

µxν = gµνdxµ

dτ

dxν

dτ

is conserved along curves of extremal proper time:d

dτ. = 0

It better be! 4-velocity uµ =dxµ

dτ, gµνu

µuν = −1

Example: Schwarzschild metric in Schwarzschild coords.:

ds2 = −f dt2 + f−1dr2 + r2dθ2 + r2 sin2 θ dφ2 , f = 1− 2M

r, M = const

⇒ L = f t2 − f−1r2 − r2θ2 − r2 sin2 θ φ2

EL for t(τ):d

dτ

(2f t)= 0 ⇒ d2t

dτ 2+ f−1 df

drt r = 0

⇒ Γttr = Γtrt =df/dr

2f, Γtµν = 0 otherwise

cf. Example sheet 1

4 COVARIANT DERIVATIVE 33

4 Covariant derivative

4.1 Introduction

Physical laws involve derivatives.

For functions we have:∂f

∂xµare the components of the gradient df

Vectors and tensors: This does not work. We cannot take the difference

between vectors at different points: U ∈ Tp(M), V ∈ Tq(M)

→ Covariant derivative ∇ on manifoldM

Def.: “Covariant derivative ∇” := map from two smooth

vectorfields X, Z to a smooth vectorfield ∇XZ with

(1) ∇fX+gY Z = f ∇XZ + g∇Y Z , f, g functions

(2) ∇X(Y +Z) = ∇XY +∇XZ

(3) ∇X(fY ) = f ∇XY + (∇Xf)Y “Leibniz”; ∇Xf := X(f)

Comments: we can view ∇Y : Tp(M)→ Tp(M) , X 7→ ∇XY

or ∇Y : T ∗p (M)× Tp(M)→ R , (η,X) 7→ η(∇XY ) ;

(11

)tensor

Def.: The(11

)tensor ∇Y is the covariant derivative of Y :

Notation: (∇Y )ab = ∇bYa = Y a

;b

Comment: • for a function f : ∇f : X 7→ ∇Xf = X(f) is a(01

)tensor

• we cannot view ∇ : (X,Y ) 7→ ∇XY as a(12

)tensor field:

not linear in Y .

Def.: Let eµ be a basis. We define the

“connection components” Γµνρ: ∇ρeν := ∇eρeν = Γµνρeµ

Example: The Christoffel symbols are one connection:

the “Levi-Civita” connection in a coord. basis; cf. below.


Comment: For a vectorfield V and a coord. basis,

we can define T µν := ∂νVµ =

∂V µ

∂xν.

This is not chart independent and, hence, not a tensor.

We are missing the variation of the basis vectors!

For an arbitrary basis eµ write:

X = Xµeµ , Y = Y µ

eµ

⇒ ∇XY = ∇X(Y µeµ) = X(Y µ)eµ + Y µ∇Xeµ

= Xνeν(Y

µ) eµ + Y µ∇Xνeνeµ

= Xνeν(Y

µ) eµ + Y µXν ∇νeµ︸︷︷︸

=Γρµνeρ

= Xν(

eν(Yµ) + ΓµρνY

ρ)

eµ

⇒ (∇XY )µ = Xνeν(Y

µ) + ΓµρνYρXν

∣∣∣∣X arbitrary

⇒ (∇Y )µν = ∇νYµ = Y µ

;ν = eν(Yµ) + ΓµρνY

ρ

Coord. basis ⇒ ∇νYµ = ∂νY

µ + ΓµρνYρ

Change of basis

eµ = (A−1)νµeν

⇒ . . .⇒ Γµνρ = Aµτ (A−1)λν(A

−1)σρΓτλσ + Aµτ (A

−1)σρ eσ((A−1)τ ν

)

︸︷︷︸

independent of Γ !

∣∣∣ Ex. sheet 2

⇒ Difference of 2 connections Γµνρ − Γµνρ transforms as tensor


Covariant derivative of tensors

Obtained from Leibniz rule; (r, s) tensor T 7→ ∇T is (r, s+ 1) tensor

E.g. 1-form:(∇Xη

)(Y ) := ∇X

(η(Y )

)− η(∇XY )

∇η is a (0, 2) tensor since:(∇Xη

)(Y ) = ∇X(ηµY

µ)− ηµ(∇XY )µ

= X(ηµ)Yµ + ηµX(Y µ)− ηµ

(Xν

eν(Yµ)

︸︷︷︸

= 0

+ΓµρνYρXν

)

= Xνeν(ηµ)Y

µ − ΓµρνηµYρXν

=(eν(ηρ)− Γµρνηµ

)XνY ρ is linear in X , Y

Components: ηµ ;ν = ∇µηµ = eν(ηµ)− Γρµνηρ

= ∂νηµ − ΓρµνηρCoord. basis:

Covariant derivative of (r, s) tensor:

∇ρTµ1...µr

ν1...ν2 = ∂ρTµ1...µr

ν1...νs + Γµ1σρ Tσµ2...µr

ν1...ν2 + . . .+ Γµrσρ Tµ1...µr−1σ

ν1...νs

−Γσν1ρ Tµ1...µr

σν2...νs − . . .− Γσνsρ Tµ1...µr

ν1...νs−1σ

Higher derivatives

f,µν = ∂ν∂µf or Xa;bc = ∇c∇bX

a

Note order of indices! Derivatives sometimes commute, sometimes not.

E.g. ∂ν∂µf = ∂µ∂νf

but ∇ν∇µf = ∇ν∂µf = ∂ν∂µf − Γρµν ∂ρf

= ∇µ∇νf − Γρµν ∂ρf + Γρνµ ∂ρf

= ∇µ∇νf − 2Γρ[µν] ∂ρf

Def.: “Torsion tensor” Tµνλ := Γλµν − Γλνµ

Γ is torsion free :⇔ Γλ[µν] = 0

Lemma: Γ torsion free, X, Y vector fields ⇒ ∇XY −∇Y X = [X,Y ]


Proof: Coord. basis

⇒ Xν∇νYµ−Y ν∇νX

µ = Xν∂νYµ +XνΓµρνY

ρ − Y ν∂νXµ − Y νΓµρνX

ρ

= [X, Y ]µ + 2Γµ[ρν]XνY ρ = [X, Y ]µ

Note: Even with torsion-free connection, 2nd cov. derivs. of

tensor fields generally do not commute.

4.2 The Levi-Civita connection

A metric singles out a preferred connection.

Fundamental theorem of Riemannian geometry:

On a manifoldM with metric g, there exists a unique,

torsion-free connection with ∇g = 0: The “Levi-Civita connection”

Proof: 1) Uniqueness

Let ∇ be a Levi-Civita connection, X, Y , Z vector fields

⇒ X(g(Y ,Z)

)= ∇X

(g(Y ,Z)

)= g

(∇XY ,Z

)+ g(Y ,∇XZ

)+ 0

Z(g(X,Y )

)= ∇Z

(g(X,Y )

)= g

(∇ZX,Y

)+ g(X,∇ZY

)+ 0

Y(g(Z,X)

)= ∇Y

(g(Z,X)

)= g

(∇Y Z,X

)+ g(Z,∇Y X

)+ 0

⇒X(g(Y ,Z)

)+ Y

(g(Z,X)

)−Z

(g(X,Y )

)

= g(∇XY +∇Y X,Z

)− g

(∇ZX −∇XZ,Y

)+ g(∇Y Z −∇ZY ,X

)

Torsion free: ∇XY −∇Y X = [X,Y ]; permute X, Y , Z

⇒X(g(Y ,Z)

)+ Y

(g(Z,X)

)−Z

(g(X,Y )

)

= 2g(∇XY ,Z

)− g([X,Y ],Z

)− g

([Z,X],Y

)+ g([Y ,Z],X

)

⇒ g(∇XY ,Z

)=

1

2

X(g(Y ,Z)

)+ Y

(g(Z,X)

)−Z

(g(X,Y )

)

+g([X,Y ],Z

)+ g([Z,X],Y

)− g([Y ,Z],X

)(∗)

g non-degenerate ⇒ unique expression for ∇XY


2) Existence: Is ∇X thus defined a connection?

Check (1) of the definition of the covariant derivative.

Let f be a function; use (∗) with X → fX

⇒ g(∇fXY ,Z

)=

1

2

fX

(g(Y ,Z)

)+ Y

(f g(Z,X)

)−Z

(f g(X ,Y )

)

+g([fX,Y ],Z

)+ g([Z, fX],Y

)− f g

([Y ,Z],X

)

=1

2

fX(g(Y ,Z)

)+ f Y

(g(Z,X)

)+Y (f) g(Z,X)− f Z

(g(X,Y )

)

−Z(f) g(X,Y )

+ f g([X,Y ],Z

)−Y (f) g(X,Z)

+f g([Z,X],Y

)+Z(f) g(X ,Y )

− f g([Y ,Z],X

)

=f

2

X(g(Y ,Z)

)+ Y

(g(Z,X)

)−Z

(g(X ,Y )

)

+g([X,Y ],Z

)+ g([Z,X],Y

)− g

([Y ,Z],X

)

= f g(∇XY ,Z

)= g

(f ∇XY ,Z

)

⇒ g(∇fXY − f ∇XY ,Z

)= 0 ∀ Z

g non-degenerate ⇒ ∇fXY = f ∇XY

(2), (3) of the definition of the cov. deriv. can be shown similarly.

Components of Levi-Civita connection in coord. basis

Use (∗) with [eµ, eν ] = 0

⇒ g(∇ρeν︸︷︷︸

=Γµνρeµ

, eσ)=

1

2

[eρ(gνσ) + eν(gσρ)− eσ(gρν)

]

⇒ g(Γµνρeµ, eσ

)= Γµνρgµσ =

1

2

(∂ρgνσ + ∂νgσρ − ∂σgρν

)∣∣∣ · gλσ

⇒ δλµΓµνρ = Γλνρ =

1

2gλσ(∂ρgνσ + ∂νgσρ − ∂σgρν

)← Christoffel symbols!


Comment: In GR we take the Levi-Civita connection.

Different connection → ∆Γ which is a tensor

→ can be viewed as matter source

4.3 Geodesics

Curves extremizing proper time:d2xµ

dτ 2+ Γµνρ

(x(τ)

) dxν

dτ

dxρ

dτ= 0 (∗)

here: τ = proper time along curve

Xµ =dxµ

dτ= tangent vector along curve

Let’s extend X to be a vectorfield in a neighbourhood of curve.

⇒ d2xµ

dτ 2=dXµ

(x(τ)

)

dτ=dxν

dτ

∂Xµ

∂xν= Xν∂νX

µ

LHS independent of the extension ⇒ RHS too.

(∗)⇒ Xν(∂νX

µ + ΓµνρXρ)= Xν∇νX

µ = 0 or ∇XX = 0 .

We derived this for the Levi-Civita connection but define for any connection:

Def.: “affinely parametrized geodesic”

:= integral curve of vector field X with ∇XX = 0

Comment: Let u be another parameter of the curve

such that τ = τ(u) ,dτ

du> 0 .

⇒ tangent vector now: Y = hX with h :=dτ

du

⇒∇Y Y = ∇hX(hX) = h∇X(hX) = h2∇XX︸︷︷︸

=0

+X(h) hX =dh

dτY

⇒∇Y Y =dh

dτY describes the same geodesic.

Unlessdh

dτ= 0 , it is not affinely parameterized.

u is also an affine parameter ⇔ h constant ⇔ u = aτ + b , a, b = const

⇒ 2 parameter family of affine parameters


Note: • For any connection, we can write the geodesic eq. (∗) for some affine parameter.

• Curves of extremal proper time are timelike geodesics.

We can also define spacelike geodesics through (∗).

Then τ is not proper time but arc length often denoted by s.

Theorem: LetM be a manifold with connection, p ∈M, Xp ∈ Tp(M)

⇒ ∃ unique affinely parametrized geodesic with tangent vector Xp in p

Proof: Let xµ be a coord. chart in nbhd. of p, Xµp components of Xp

geodesic eq.:d2xµ

dτ 2+ Γµνρ

dxν

dτ

dxρ

dτ= 0

with initial conditions xµ(0) = xµ(p) ,dxµ

dτ

∣∣∣∣τ=0

= Xµp

This is a system of ODEs for xµ. Theory of ODEs

⇒ unique solution exists.

Note: Levi-Civita connection, ∇XX = 0 along affinely parametrized geodesic implies:

∇X

(g(X,X)

)=(∇Xg

)(X,X) + 2g

(∇XX,X

)= 0 + 0

⇒ g(X,X) const. along curve

⇒ tangent vector cannot change time, space or null character

⇒ geodesic is either time, spacelike or null

Postulate: massive (massless) particles in GR move on timelike (null) geodesics

Note: Null geodesics have no analogue of proper time or arc length,

but still affine parameters.


4.4 Normal coordinates

Def.: LetM be a manifold, Γ a connection, p ∈M.

“exponential map” := e : Tp →M , Xp 7→ q with

q := point a unit affine parameter distance along

geodesic through p with tangent Xp

Comments: 1) e can be shown to be one-to-one and onto locally,

(geodesics can cross globally)

2) The vector Xp fixes the parametrization of the geodesic:

One can show that tXp , 0 ≤ t ≤ 1 is mapped to point at

affine par. distance t along the geodesic of Xp. (∗∗)

Def.: Let eµ be a basis of Tp(M). “Normal coords. in nbhd. of p ∈M”:

chart that assigns to q = e(X) ∈M the coordinates Xµ

Note: The coords. Xµ are not fixed by the vector X .

We still have the freedom to choose a basis for Tp(M).

Lemma: In normal coordinates, Γµ(νρ) = 0 at p.

If Γ is torsion free, then Γµνρ = 0.

Proof: From (∗∗) ⇒ affinely parametrized geodesic is given by

xµ(t) = tXµp in normal coords.

⇒ geodesic eq.: 0 + Γµνρdxν

dt

dxρ

dt= ΓµνρX

νpX

ρp = 0 at p ∀X ∈ Tp(M)

⇒ Γµ(νρ) = 0

torsion free ⇒ Γµ[νρ] = 0 ⇒ Γµνρ = 0

Note: in general Γµνρ 6= 0 away from p !


Lemma: with metric, we can use the Levi-Civita connection

⇒ in normal coords. at p: ∂ρgµν = gµν,ρ = 0

Proof: Γρµν = 0 ⇒ 2gσρΓρµν = ∂νgσµ + ∂µgνσ − ∂σgµν = 0

symmetrize on σ, µ, add ⇒ ∂νgσµ = 0

Note: Again valid only at p !

In general we cannot make ∂νgσµ vanish away from p.

Lemma: LetM be a manifold with metric gµν and torsion free connection.

⇒ we can choose normal coords. such that at p:

∂ρgµν = 0 , gµν = ηµν (or δµν in Riemannian case)

Proof: Choose an orthonormal basis eµ for Tp(M).

Let X be a vector field.

⇒ at p: X = X1e1 + . . .+Xn

en defines normal coords. xµ = Xµ

Consider vector∂

∂x1⇒ its integral curve is xµ(t) = (t, 0, . . . , 0)

becaused

dt=dxµ

dt

∂

∂xµ= δµ1

∂

∂xµ=

∂

∂x1

The components of the tangent vector to the curve xµ(t) = (t, 0, . . . , 0)

are also:dxµ

dt= (1, 0, . . . , 0)

⇒ The tangent vector is e1 ⇒ e1 =∂

∂x1

Likewise: eµ =∂

∂xµ

→

∂

∂xµ

defines a coordinate orthonormal basis.

Summary: Locally, we can choose coordinates such that the metric is ηµν = diag(−1, 1, 1, 1)and its first derivatives vanish.

Def.: “local inertial frame at p ∈M′′ := normal coord. chart with these properties

5 PHYSICAL LAWS IN CURVED SPACETIME 42

5 Physical laws in curved spacetime

5.1 Covariance

“general covariance”: Physical laws should be independent of

the choice of charts and basis.

“special covariance” in special relativity: laws independent of inertial frame

Recipe for converting SR laws → GR laws

1) ηµν → gµν Minkowski → curved metric

2) ∂ → ∇ partial → covariant derivs.

3) µ, ν, . . .→ a, b, . . . coord. indices → abstract indices

Examples:

1) Let xµ be inertial frame coords., ηµν the Minkowski metric.

⇒ scalar wave eq.: ηµν∂µ∂νφ = 0 in SR

→ gab∇a∇bφ = ∇a∇aφ = φ;aa = 0

2) Electromagnetic field in SR:

Fµν = F[µν] with F0i = −Ei , Fij = ǫijkBk , (i, j, k = 1 . . . 3)

vacuum Maxwell eqs.: ηµν∂µFνρ = 0 , ∂[µFνρ] = 0

→ in GR: gab∇aFbc = 0 , ∇[aFbc] = 0

Lorentz force in SR:d2xµ

dτ 2=

q

mηµνFνρ

dxρ

dτ︸︷︷︸

; τ = proper time

= uρ = 4-velocity

→ in GR: ub∇bua =

q

mgabFbcu

c


Comment: This procedure satisfies the EP. In a local inertial frame:

Γµνρ∣∣p= 0 , gµν

∣∣p= ηµν

⇒ ∇→ ∂ , so in a LIF we have SR.

But: The step SR→ GR is not unique.

E.g. we can add curvature terms to the GR eqs.

Such terms are zero in SR (see below).

Ultimate test: experiment.

5.2 Energy momentum tensor

Energy, momentum, mass source gravity. Ho do we describe them in GR?

Particles

1) in SR: Associate rest mass with particle

⇒ 4-momentum P µ = muµ = (E, P i) in this frame

4-velocity of observer in particle’s rest frame: vµ = (1, 0, 0, 0)

particle energy measured by this observer: E = −ηµνvµP ν

particle’s rest mass: ηµνPµP ν = −E2 + ~p2 = −m2 ; note: c = 1

2) in GR: EP ⇒ P a = mua ⇒ gabPaP b = −m2

Particle energy measured by observer: E = −gab(p) va(p)P b(p)

works only if both are at p

An observer at p ∈M cannot measure the energy of a particle at q


electromagnetic field

1) pre-relativistic notation, Cartesian coordinates:

energy density: ǫ =1

8π(EiEi +BiBi)

momentum density, energy flux: Si =14πǫijkEjBk “Poynting vector”

Maxwell eqs. ⇒ ∂ǫ

∂t+ ∂iSi = 0

stress tensor: tij =1

4π

[1

2(EkEk +BkBk)δij − EiEj − BiBj

]

conservation law:∂Si∂t

+ ∂jtij = 0

Force exerted on surface element dA with normal ni: tijnjdA

2) Special relativity:

energy momentum tensor (= stress tensor = stress-energy tensor) in IF:

Tµν =1

4π

(

FµρFνρ − 1

4F ρσFρσηµν

)

= Tνµ

T00 = ǫ , T0i = −Si , Tij = tij ; from 1)

Conservation: ∂µTµν = ηµσ∂σTµν = 0

3) GR: we define by covariance:

Tab =1

4π

(

FacFbc − 1

4F cdFcdgab

)

Maxwell eqs.: ∇aTab = 0 ; cf. example sheet 2

Postulate: In GR, continuous matter is described by a conserved, symmetric

(0, 2) tensor which contains the information about the matter’s

energy, momentum and stress.

The energy momentum tensor is conserved: ∇aTab = 0.


Comment: Let O be an observer with 4-velocity ua.

Consider a LIF at p where O is at rest.

Choose orthonormal basis eµ at p aligned with the coord. axes of this LIF.

⇒ ea0 = ua , spatial basis vectors eai , i = 1, 2, 3

EP ⇒ ǫ = T00 = Tabea0eb0 = Tabu

aub = energy density at p measured by O

Si = −T0i = momentum density

ja = −T abub = (ǫ, Si) in this basis = energy momentum current

tij = Tij = stress tensor as measured by O

Comment: Consider an IF in Minkowski spacetime.

local conservation ∂µTµνintegration

−−−−−−−−→ global conservation

E.g.:∂ǫ

∂t+ ∂iSi = 0 ⇒ d

dt

∫

V

ǫ dV = −∫

∂V

~S · ~n dA

in GR: This is not possible! The grav. field contains energy,

but there is no invariant definition for it.

Newtonian analogue − 1

8π(~∇φ)2 does not work because

metric derivatives vanish in normal coordinates.

⇒ energy only defined for global spacetime or special cases, e.g. horizons

Example: A perfect fluid is matter described by a 4-velocity field ua and

functions ρ, p such that Tab = (ρ+ p)uaub + pgab .

ρ, p = energy density, pressure measured by observer co-moving with fluid

One can show: 1) Tabuaub = ρ

2) ∇aTab = 0 ⇔ ua∇aρ+ (ρ+ p)∇aua = 0

∧ (ρ+ p)ub∇bua = −(gab + uaub)∇bp

These are GR’s version of the Euler eqs. and mass conservation.

Note: p = 0⇒ fluid moves on geodesics.

6 CURVATURE 46

6 Curvature

6.1 Parallel transport

A connection gives us a notion of “a tensor that does not change along a curve”

Def.: Let X be tangent to a curve. A tensor is

“parallely transported/propagated along the curve” :⇔ ∇XT = 0

Comments: • The tangent of a geodesic is parallely propagated along itself.

• ∇XT = 0 determines T uniquely along the curve:

in coords. (xµ) the curve is xµ(t)

⇒ Xσ∇σTµν = Xσ∂σT

µν + ΓµρσT

ρνX

σ − ΓρνσTµρX

σ

=d

dtT µν + ΓµρσT

ρνX

σ − ΓρνσTµρX

σ = 0

ODE theory ⇒ unique solution for all T µν

• q ∈ M, q 6= p : parallel transport T along a curve from p to q

→ isomorphism between tensors at p, q

• Euclidean space or Minkowski in Cartesian coords.

⇒ Γµνρ = 0 ⇒ d

dtT µν = 0

⇒ parallel trapo. leaves tensor components constant

⇒ parallel trapo. is independent of the curve chosen!

This is not the case in GR!

6.2 The Riemann tensor

Def.: The Riemann curvature tensor Rabcd is defined such that

for VFsX, Y , Z: RabcdZ

bXcY d =(R(X,Y )Z)a with

R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z

6 CURVATURE 47

Linearity in X, Y , Z: Let f be a function.

(1) R(fX,Y )Z = ∇fX∇Y Z −∇Y∇fXZ −∇[fX,Y ]Z

= f∇X∇Y Z −∇Y (f∇XZ)−∇f [X,Y ]−Y (f)XZ

= f∇X∇Y Z − f∇Y∇XZ − Y (f)∇XZ −∇f [X,Y ]Z +∇Y (f)XZ

= f∇X∇Y Z − f∇Y∇XZ−Y (f)∇XZ

− f∇[X,Y ]Z+Y (f)∇XZ

= f R(X,Y )Z

(2) R(X,Y )Z = −R(Y ,X)Z ⇒ linear in Y too

(3)R(X,Y ) (fZ) = ∇X∇Y (fZ)−∇Y∇X(fZ)−∇[X,Y ](fZ)

= ∇X

(f ∇Y Z + Y (f)Z

)−∇Y

(f ∇XZ +X(f)Z

)− f∇[X,Y ]Z − [X,Y ](f)Z

= f ∇X∇Y Z +X(f)∇Y Z + Y (f)∇XZ

+X(Y (f)

)Z

−f∇Y∇XZ − Y (f)∇XZ

−X(f)∇Y Z − Y(X(f)

)Z

−f ∇[X,Y ]Z − [X,Y ](f)Z

= f R(X,Y )Z

Coord. basis

eµ =∂

∂xµ

⇒ [eµ, eν ] = 0 ; ∇µ := ∇eµ

⇒ R(eρ, eσ)eν = ∇ρ∇σeν −∇σ∇ρeν

= ∇ρ(Γτνσeτ )−∇σ(Γ

τνρeτ )

= ∂ρΓµνσeµ + ΓτνσΓ

µτρeµ − ∂σΓµνρeµ − ΓτνρΓ

µτσeµ

⇒ Rµνρσ = ∂ρΓ

µνσ − ∂σΓµνρ + ΓτνσΓ

µτρ − ΓτνρΓ

µτσ (∗)

Comment: Rµνρσ = 0 in Minkowski or Euclidean:

We can choose coords. such that Γµνρ = 0 everywhere.

Def.: “Ricci tensor” Rab := Rcacb

6 CURVATURE 48

Comments: 2nd cov. derivs. of functions commute ⇔ no torsion

2nd cov. derivs. of tensors do not commute even if torsion = 0

e.g. one can show: (∇c∇d −∇d∇c)Za = Ra

bcdZb “Ricci identity”

6.3 Parallel transport and curvature

Let X, Y be VFs with: lin. indep. everywhere and [X,Y ] = 0; let torsion = 0

⇒ we can choose coords. (s, t, . . .) such that X =∂

∂s, Y =

∂

∂t

Let p, q, r, u ∈M along integral curves of X, Y with coords.

(0, . . . , 0), (δs, 0, . . .), (δs, δt, 0, . . .), (0, δt, 0, . . .)

X

X

YY

p (0,0) δq ( s,0)

r ( s, t)δ δu (0, t)δ

Let Zp ∈ Tp(M) , parallel trapo Z along pqrup

to get Z ′p ∈ Tp(M)

⇒ limδ→0

(Z ′p −Zp)

a

δs δt= (Ra

bcdZbY cXd)p

Proof:

Let Zp ∈ Tp(M) , (xµ) be normal coords. at p.

s, t are now “only” parameters along the curves.

We assume δs, δt are small and δt = a δs for a = const.

6 CURVATURE 49

(1) p→ q: curve with tangent X and parameter s

parallel transport ∇XZ = 0

⇒ Xσ∇σZµ = Xσ ∂

∂xσZµ + ΓµρσZ

ρXσ =dZµ

ds+ ΓµρσZ

ρXσ = 0

⇒ dZµ

ds= −ΓµρσZρXσ

⇒ d2Zµ

ds2= −Xλ∂λ(Γ

µρσZ

ρXσ)∣∣∣ X = Xµ ∂

∂xµ=

d

ds

Taylor expand around p and use Γµρσ∣∣p= 0

⇒ Zµq − Zµ

p =

(dZµ

ds

)

p

δs+1

2

(d2Zµ

ds2

)

p

δs2 +O(δs3)

= −12

(XλZρXσ∂λΓ

µρσ

)

pδs2 +O(δs3)

(2) q → r: Zµr − Zµ

q =

(dZµ

dt

)

q

δt+1

2

(d2Zµ

dt2

)

q

δt2 +O(δt3)

= −(ΓµρσZ

ρY σ)

qδt

︸︷︷︸

−12

[Y λ∂λ(Γ

µρσZ

ρY σ)]

qδt2 +O(δt3)

=[(ΓµρσZ

ρY σ)p +(Xλ∂λ(Γ

µρσZ

ρY σ))

pδs+O(δs2)

]δt

=[0 + (ZρY σXλ∂λΓ

µρσ)pδs+O(δs2)

]δt

⇒ Zµr − Zµ

q = −[(ZρY σXλ∂λΓ

µρσ)pδs+O(δs2)

]δt

−12

[ (Y λ∂λ(Γ

µρσZ

ρY σ))

p︸︷︷︸

+O(δs)]δt2 +O(δt3)

= (ZρY σY λ∂λΓµρσ)p

⇒(Zµr − Zµ

p

)

pqr= −1

2(∂λΓ

µρσ)[Zρ(XσXλδs2 + Y σY λδt2 + 2Y σXλδs δt

)]

p+O(δt3)

We obtain (Zµr − Zµ

p )pur by simply interchanging X ↔ Y , s↔ t

⇒(Zµr − Zµ

p

)

pur= −1

2(∂λΓ

µρσ)[Zρ(Y σY λδt2 +XσXλδs2 + 2XσY λδt δs

)]

p+O(δs3)

6 CURVATURE 50

⇒ Z ′µp −Zµ

p = (Zµr − Zµ

p )pqr − (Zµr − Zµ

p )pur = −[(Y σXλ −XσY λ)(∂λΓ

µρσ)]

pZρ δs δt+O(δ3)

=[XσY λZρ (∂λΓ

µρσ − ∂σΓµρλ)

︸︷︷︸

]

p+O(δ3) =

(Rµ

ρλσZρY λXσ

)

p+O(δ3)

(∗)= Rµ

ρλσ in normal coords: (Γαβγ)p = 0

Conclusion: Curvature measures the change of vectors under parallel transport

along closed curves or, equivalently, the path (in)dependence of par. trapo.

6.4 Symmetries of the Riemann tensor

(i) Rabcd = −Ra

bdc ⇔ Rab(cd) = 0 by def.

Torsion = 0, let p ∈M, (xµ) normal coords. Then:

(ii) Γµνρ = 0 at p, Γµ[νρ] = 0 everywhere

⇒ Rµνρσ = ∂ρΓ

µνσ − ∂σΓµνρ

∣∣∣ antisymmetrize on νρσ

⇒ Rµ[νρσ] = 0 ⇒ Ra

[bcd] = 0

(iii) ∇τRµνρσ = ∂τR

µνρσ

∣∣∣ “∂R = ∂∂Γ − Γ ∂Γ = ∂∂Γ”

= ∂τ∂ρΓµνσ − ∂τ∂σΓµνρ

∣∣∣ antisymmetrize on ρστ

⇒ Rµν[ρσ;τ ] = 0 “Bianchi identity”

⇒ Rab[cd;e] = 0 tensorial equation !

6 CURVATURE 51

6.5 Geodesic deviation

Goal: quantify relative acceleration of geodesics

Def.: Let (M,Γ) be a manifold with connection.

“1-parameter family of geodesics” := a map

γ : I × I ′ →M with I, I ′ ⊂ R, open and

(i) for fixed s, γ(s, t) is a geodesic with affine par. t

(ii) locally, (s, t) 7→ γ(s, t) is smooth, 1-to-1 has smooth inverse

⇒ the family of geodesics forms a 2-dim. surface Σ ⊂M

Let T be the tangent vector to γ(s = const, t) and S to γ(s, t = const)

In coords. (xµ): Sµ =∂xµ

∂s

⇒ xµ(s+ δs, t) = xµ(s, t) + δs Sµ(s, t) +O(δs2)

⇒ δsS points from one geodesic to a nearby one: “deviation vector”

⇒ “relative velocity” of nearby geodesics: ∇T (δsS) = δs∇TS

⇒ “relative acceleration” of nearby geodesics: δs∇T∇TSt=

s=

const

const

T

S

Geodesic deviation: ∇T∇TS = R(T ,S)T

⇔ T c∇c(Tb∇bS

a) = RabcdT

bT cSd

Proof: Use coords. (s, t) on Σ and extend to (s, t, . . .) in nbhd. of Σ

⇒ S =∂

∂s, T =

∂

∂t⇒ [S,T ] = 0

No torsion ⇒ ∇TS −∇ST = [T ,S] = 0

⇒ ∇T∇TS = ∇T∇ST = ∇S∇TT︸︷︷︸

+R(T ,S)T

= 0 geodesic!

6 CURVATURE 52

Comments: • Rabcd measures geodesic deviation; manifestation of curvature.

• Rabcd = 0⇔ relative acceleration = 0 for all families of geodesics.

• Tidal forces arise from geodesic deviation.

6.6 Curvature of the Levi-Civita connection

From now on: A manifold is assumed to have a metric and

and the connection is the Levi-Civita one unless stated otherwise.

Note: Rabcd = gaeRebcd

Def.: “Ricci scalar” R := gabRab

“Einstein tensor” Gab := Rab −1

2Rgab

Propositions: (1) Rabcd = Rcdab

(⇒ R(ab)cd = Rcd(ab) = 0 ⇒ Rbacd = −Rabcd

)

(2) Rab = Rba

(3) ∇aGab = 0 “contracted Bianchi identities”

Proof: (1) Let p ∈M, use normal coords. at p ⇒ ∂µgνρ = 0

⇒ 0 = ∂µδνρ = ∂µ(g

νσgσρ) = gσρ∂µgνσ

∣∣∣ · gρτ

⇒ ∂µgντ = 0

⇒ ∂ρΓτνσ =

1

2gτµ(∂ρ∂σgµν + ∂ρ∂νgσµ − ∂ρ∂µgνσ

)

⇒ Rµνρσ =1

2

(∂ρ∂νgσµ + ∂σ∂µgνρ − ∂σ∂νgρµ − ∂ρ∂µgνσ

)+ “ΓΓ− ΓΓ”︸︷︷︸

=0

= Rρσµν because gαβ symmetric, ∂α∂β commute

(2) Rab = gcdRdacb = gcdRcbda = Rba

(3) Example sheet.

6 CURVATURE 53

6.7 Einstein’s equation

Postulates of GR

(1) Spacetime is a 4-dim. Lorentzian manifold with metric and Levi-Civita connection.

(2) Free particles follow timelike or null geodesics.

(3) Energy, momentum and stress of matter is described by a symmetric,

conserved tensor Tab : ∇aTab = 0 .

(4) Curvature is related to matter by the Einstein eqs.

Gab = Rab −1

2gabR =

8πG

c4Tab ; G = Newton’s constant

Comments:

(i) Simplest relation between curvature and energy-momentum is linear.

→ Einstein’s first guess: Rab = κTab ; κ = const

But: ∇aGab = ∇aRab −1

2gab∇aR = 0− 1

2gab∇aR

∣∣∣ because ∇aTab = 0

!= 0 ⇒ ∇aR = 0 ⇒ ∇aT = 0

not satisfactory since T = 0 outside and T 6= 0 inside matter

Solution: replace Rab with Gab ← “contracted Bianchi Id.”

κ follows from Newtonian limit; cf. below.

(ii) Vacuum ⇒ Gab = Rab −1

2gabR = 0

∣∣∣ · gab

⇒ R = 0 ⇒ Rab = 0

(iii) The geodesic postulate can be shown to follow from ∇aTab = 0

(iv) Gab =8πG

c4Tab are 10 coupled, non-linear PDEs → tough to solve

6 CURVATURE 54

Theorem: (Lovelock 1972) Let Hab be a symmetric tensor with

(i) in any chart Hµν = Hµν(gµν , ∂ρgµν , ∂σ∂ρgµν) at every p ∈M

(ii) ∇aHab = 0

(iii) Hµν linear in ∂σ∂ρgµν

⇒ ∃α,β∈R Hab = αGab + βgab

⇒ we can modify Einstein’s eq.: Gab + Λgab = 8πTab

→ Cosmological constant: Λ ; |Λ|−1/2 ≈ 109 light years (from observations);

Λ can be regarded as a perfect fluid with ρ = −p = Λ c4

8πG: “dark energy”

6.8 Units

In GR we often set G = 1, c = 1

G = 6.67× 10−11 m3

kg s2, c = 3× 108

m

s

⇒ 1 s = 3× 108 m

∧ 1 kg = 0.74× 10−27 m

E.g. M⊙ ≈ 1.48 km

7 DIFFEOMORPHISMS AND LIE DERIVATIVE 55

7 Diffeomorphisms and Lie derivative

7.1 Maps between manifolds

Def.: Let M, N be differentiable manifolds of dimension m, n respectively.

A function φ :M→N is smooth

:⇔ ψA φ ψ−1α : Rm → R

n is smooth ∀ charts ψα onM, ψA on N

Def.: Let φ :M→N , f : N → R be smooth. The “pull-back of f by φ” is

φ∗(f) :M→ R , p 7→ φ∗(f) (p) := f(φ(p)

)

Def.: The “push-forward of a curve λ : I ⊂ R→M” is

φ λ : I ⊂ R→ N , t 7→ φ(λ(t)

)

Def.: Let p ∈M , X ∈ Tp(M) be the tangent vector of λ : I ⊂ R→M

The “push-forward of X by φ” is

φ∗(X) ∈ Tφ(p)(N ) defined as tangent vector of φ λX

φ∗(X)

φ(p)

M

N

φ

p

φ λλ

Lemma: Let X ∈ Tp(M), f : N → R

⇒(φ∗(X)

)(f) = X

(φ∗(f)

)

Proof: Let λ(0) = p

⇒(φ∗(X)

)(f)∣∣∣φ(p)

=

[d

dt

(f (φ λ)

)(t)

]∣∣∣∣t=0

=

[d

dt

((f φ) λ

)(t)

]∣∣∣∣t=0

= X(φ∗(f)

)∣∣p

Def.: Let φ :M→N be smooth, p ∈M, η ∈ T ∗φ(p)(N ).

The “pull-back of η by φ” is

φ∗(η) ∈ T ∗p (M) ,

(φ∗(η)

)(X) = η

(φ∗(X)

)∀X ∈ Tp(M)


Lemma: Let f : N → R ⇒ df ∈ T ∗φ(p)(N ).

Then φ∗(df) ∈ T ∗p (M) is φ∗(df) = d

(φ∗(f)

)

Proof: Let X ∈ Tp(M)

⇒(φ∗(df)

)(X) = df

(φ∗(X)

)=(φ∗(X)

)(f) = X

(φ∗(f)

)=[d(φ∗(f)

)](X)

Components

Let xµ be coords. onM, yα coords. on N , µ = 1 . . .dim(M), α = 1 . . .dim(N )

⇒ φ :M→N defines a map xµ 7→ yα(xµ)

One can show: for a vector X ∈ Tp(M) :(φ∗(X)

)α=∂yα

∂xµ

∣∣∣∣p

Xµ

for a 1-form η ∈ T ∗φ(p)(N ) :

(φ∗(η)

)

µ=∂yα

∂xµ

∣∣∣∣p

ηα

Comments:

• p ∈ M was arbitrary

⇒ push-forward applies to vector fields, pull-back to covector fields

• pull-back of(0s

)tensor S:

(φ∗(S)

)(X1, . . . , Xs) := S

(φ∗(X1), . . . , φ∗(Xs)

)∀X1, . . . , Xs ∈ Tp(M)

push-forward of(r0

)tensor T :

(φ∗(T )

)(η1, . . . , ηr) := T

(φ∗(η1), . . . , φ

∗(ηr))∀η1, . . . , ηr ∈ T ∗

φ(p)(N )

Components:(φ∗(S)

)

µ1...µs=∂yα1

∂xµ1

∣∣∣∣p

· . . . · ∂yαs

∂xµs

∣∣∣∣p

Sα1...αs

(φ∗(T )

)α1...αr=∂yα1

∂xµ1

∣∣∣∣p

· . . . · ∂yαr

∂xµr

∣∣∣∣p

T µ1...µr

Example: Let M = S2 (unit sphere), N = R3, xµ = (θ, φ) spherical coords. on S2

φ :M→N , p(θ, φ) 7→ yα = (sin θ cosφ, sin θ sinφ, cos θ) ∈ R3.

Let g be the Euclidean metric on R3, gαβ = δαβ in coords. (x, y, z)

⇒ . . .⇒ The pull-back of g onto S2 is: (φ∗g)µν = diag(1, sin2 θ) .


7.2 Diffeomorphisms, Lie derivative

Def.: φ :M→N is a “diffeomorphism” (dfm.)

:⇔ φ is 1-to-1, onto, smooth, and has a

smooth inverse. M, N must have the same dimension.

with a dfm., we have:

Def.: Let φ :M→N be a dfm., T a(rs

)tensor onM.

The “push-forward of T under φ is the(rs

)tensor on N :

φ∗(T )(η1, . . . , ηr, X1, . . . , Xs)

:= T(φ∗(η1), . . . , φ

∗(ηr), (φ−1)∗(X1), . . . , (φ

−1)∗(Xs))

∀ηi ∈ T ∗φ(p)(N ), X i ∈ Tφ(p)(N )

One can show:

1) Push-forward commutes with contraction and outer product.

2) Components for(11

)tensor in coord. basis:

[(φ∗(T )

)µν

]

φ(p)=∂yµ

∂xρ

∣∣∣∣p

∂xσ

∂yν

∣∣∣∣p

(T ρσ)p (∗)

generalizes obviously for(rs

)tensors

Comments: 1) pull-back of(rs

)tensors can be defined likewise ⇒ φ∗ = (φ−1)∗

2) We took “active” viewpoint: φ : p 7→ φ(p), 2 manifolds

“passive interpretation”:

pull coords. yµ back from N toM

⇒ simply 2 coord. charts xµ, yµ onM

⇒ (∗) becomes the ordinary tensor transformation law

p

M

xµyµ

N

φ(p)

xµ, yµ

pM


Def.: Let φ :M→N be a dfm., ∇ a covariant deriv. onM,

X a vector, T a tensor on N .

⇒ The push-forward of ∇ is the cov. deriv. ∇ on N defined by

∇XT := φ∗

[∇φ∗(X)

(φ∗(T )

)]

One can show (Example sheet 3):

(1) ∇ satisfies the properties of a cov. deriv.

(2) The Riemann tensor of ∇ is the push-forward of Riemann(∇)

(3) Let ∇ be the cov. deriv. of the Levi-Civita connection of g onM

⇒ ∇ is that of the Levi-Civita connection of φ∗(g) on N

Diffeomorphism invariance

We defined a spacetime as a pair (M, g).

Let’s add matter fields F , . . . → (M, g,F , . . .)

2 models (M, g,F , . . .), (M′, g′,F ′, . . .) are taken to be equivalent if

∃ dfm. φ :M→M′ which carries g, F , . . . to g′, F ′, . . . :

g′ = φ∗g, F ′ = φ∗F , . . .

active-passive equivalence ⇒ the models just differ by a coord. trafo.

⇒ A spacetime is really an equivalence class of all equivalent (M′, g′, F ′, . . .)

Consequences: 1) Einstein’s eqs. will not predict all 10 metric components!

2) Physical statements in GR must be diffeomorphism invariant.

3) This is the gauge freedom of GR.

Examples: 1) “Two geodesics intersect at xµ = (. . .)” is not gauge invariant

2) Consider a geodesic intersected exactly once by

each of two other geodesics.

⇒ The proper time along the geodesic between

the intersections is gauge invariant.


Lie derivatives, symmetries

Push-forward and pull-back provide a way to compare tensors at different p, q ∈M

Def.: A dfm. φ :M→M is a “symmetry transformation of a tensor field T ”

:⇔ φ∗(T ) = T everywhere.

“isometry” := a symmetry trafo. of the metric

Def.: Let X be a VF on a manifoldM. Let φt :M→M, p 7→ q such that

q := point a parameter distance t along the integral curve of X through p

For small enough t, φt can be shown to be a dfm.

Comments: 1) φ0 is the identity map; φs φt = φt+s; φ−t = (φt)−1.

2) If φt is a dfm. ∀t ∈ R⇒ the φt form a 1-par. Abelian group

Then we can define ∀p ∈ M the curve

λp : R→M , t 7→ φt(p).

Doing this ∀p ∈M defines a VF:

X := tangent vectors of these curves.

3) The push-forward (φt)∗ allows us to compare tensors at different points.

→ Def.: The “Lie derivative of a tensor T along a VF X at p ∈M” is

(LXT )p = limt→0

[(φ−t)∗T ]p − T p

t

Comments: • LX maps(rs

)tensor fields to

(rs

)tensor fields

• α, β const. ⇒ LX(αS + βT ) = αLXS + βLXT


Adapted coordinates

1) Let Σ be an n− 1 dim. hypersurface ofM,

X a VF that is nowhere tangent to Σ.

2) Let xi , i = 1 . . . n− 1 be coords. on Σ. Assign to q ∈M

coords. (t, xi) such that q is a parameter distance t along

the integral curve of X through xi on Σ.

→ coord. chart for sufficiently small t

Σxi

X

λ

Note: Int. curves of X have fixed (xi) and parameter t: X =∂

∂t.

The dfm. φt sends point p with (tp, xi) to q with yµ = (tp + t, xi) .

⇒ ∂yµ

∂xν= δµν

Now consider an(rs

)tensor T in these coords.:

⇒[((φt)∗T

)µ1...µrν1...νs

]

φt(p)=∂yµ1

∂xρ1. . .

∂yµr

∂xρr∂xσ1

∂yν1. . .

∂xσs

∂yνs

[T ρ1...ρrσ1...σs

]

p

=[T µ1...µrν1...νs

]

p

⇒[((φ±t)∗T

)µ1...µrν1...νs

]

p=[T µ1...µr ν1...νs

]

φ∓t(p)

⇒ at p with (tp, xi):

(LXT )µ1...µrν1...νs = limt→0

1

t

[T µ1...µr ν1...νs(tp + t, xi)− T µ1...µrν1...νs(tp, xi)

]

=

[∂

∂tT µ1...µrν1...νs(t, x

i)

]

(tp,xi)

in this chart!

It follows: Leibniz rule: LX(S ⊗ T ) = (LXS)⊗ T + S ⊗ (LXT ) ;

LX commutes with contraction


We still need a chart independent expression:

1) In this chart: LXf∗=

∂

∂tf for function f , X(f)

∗=

∂

∂tf

⇒ LXf = X(f) in any basis!

2) In our chart: (LXY )µ =∂Y µ

∂tfor VF Y ;

[X,Y ]µ =∂Y µ

∂t(because Xµ = δµ0)

⇒ LXY = [X,Y ] in any basis!

Comment: LXT depends on Xp and its derivative

⇒ L, LT are not tensors

cf. covariant deriv.: ∇XT depends only on Xp; also linear in Xp

⇒∇T is a tensor

One can show:

1) For 1-form ω: (LXω)µ = Xν∂νωµ + ων∂µXν ,

(LXω)a = Xb∇bωa + ωb∇aXb

2) For a tensor T : (LXT )α...β... = Xγ∂γTα...

β... − (∂γXα)T γ...β... − . . .+ (∂βX

γ)T α...γ... + . . .

(LXT )a...b... = Xc∇cTa...

b... − (∇cXa)T c...b... − . . .+ (∇bX

c)T a...c... + . . .

3) For metric: (LXg)µν = Xρ∂ρgµν + gµρ∂νXρ + gρν∂µX

ρ

= gµρ∇νXρ + gρν∇µX

ρ (for Levi-Civita connection)

Killing’s equation: Let φt be an isometry ∀t∈R ⇒ LXg = 0

∇aXb +∇bXa = 0 solutions X are “Killing vectors”


Note: 1) If ∃ chart with one coord. z on which gµν do not depend

⇒ ∂

∂zis Killing VF

2) Conversely, if ∃ a Killing VF

⇒ we can choose coords. such that gµν does not depend on one of them

Lemma: Let X be a Killing field and V a VF tangent

to an affinely parametrized geodesic.

d

dτ(XaV

a) = V (XaVa) = ∇V (XaV

a) = V b∇b(XaVa)

= V aV b︸︷︷︸

symm.

∇bXa︸︷︷︸

antisymm.

+XaVb∇bV

a = 0

⇒ XaVa const. along geodesic.

One can show: Tab = energy-momentum tensor, Xa = Killing VF, Ja := T abXb

⇒ ∇aJa = 0 “conserved current”

8 LINEARIZED THEORY 63

8 Linearized Theory

8.1 The linearized Einstein eqs.

Consider small deviations from Minkowski in Cart. coords.

“Background”: ManifoldM = R4, ηµν = diag(−1, 1, 1, 1)

“Perturbation”: hµν = O(ǫ)≪ 1 ⇒ gµν = ηµν + hµν

regard hµν as a tensor field on Minkowski background

2 metrics: ηµν and the “physical metric” gµν .

inverse metric: gµν = ηµν + kµν

⇒ gµνgνρ = δµρ + kµνηνρ + ηµνhνρ + kµνhνρ︸︷︷︸

=O(ǫ2)→0

!= δµρ

⇒ kµν = −ηµρηνσhρσ =: −hµν = O(ǫ)

To O(ǫ): Γµνρ =1

2ηµσ(∂ρhσν + ∂νhρσ − ∂σhνρ) ,

Rµνρσ = ηµτ(∂ρΓ

τνσ − ∂σΓτνρ

)∣∣∣ Γ · Γ = O(ǫ2)

= 12

(∂ρ∂νhµσ + ∂σ∂µhνρ − ∂ρ∂µhνσ − ∂σ∂νhµρ

)

Rµν = ∂ρ∂(µhν)ρ −1

2∂ρ∂ρhµν −

1

2∂µ∂νh

∣∣∣ h := hµµ , ∂µ := gµρ∂ρ

Gµν = ∂ρ∂(µhν)ρ −1

2∂ρ∂ρhµν −

1

2∂µ∂νh−

1

2ηµν(∂

ρ∂σhρσ − ∂ρ∂ρh) != 8πTµν

⇒ Tµν ≪ 1

Def.: “trace-reversed pert.” hµν := hµν −1

2hηµν ⇔ hµν = hµν −

1

2hηµν ,

h = hµµ = −h

⇒ . . .⇒ Gµν = −1

2∂ρ∂ρhµν + ∂ρ∂(µhν)ρ −

1

2ηµν∂

ρ∂σhρσ = 8πTµν


Gauge symmetry

Let (M, g,T ) be a spacetime, φ :M→M′ a diffeomorphism.

⇒(M′, φ∗(g), φ∗(T )

)is a physically equivalent spacetime.

We want ηµν to remain the background metric ⇒ φ ∼ O(ǫ)

Consider dfm. φt defined by integral curves of VF X ⇒ t = O(ǫ)

⇒ With ξµ = tXµ = O(ǫ) for any tensor T :

(φ−t)∗(T ) = T + tLXT +O(t2) = T + LξT +O(ǫ2)

energy momentum tensor: Tµν!= O(ǫ)⇒

((φ−t)∗T

)

µν= Tµν +O(ǫ2)

metric: (φ−t)∗(g) = g + Lξg + . . . = η + h+ Lξη +O(ǫ2)

⇒ hµν and hµν + (Lξη)µν are physically equivalent perturbations

⇒ gauge symmetry: hµν → hµν + ∂µξν + ∂νξµ , ξµ = O(ǫ)

Now choose ξµ such that ∂ν∂νξµ = −∂ν hµν⇒ ∂ν hµν → . . . = ∂ν hµν + ∂ν∂νξµ = 0

⇒ Gµν = −1

2∂ρ∂ρhµν

⇒ lin. Einstein eqs.: ∂ρ∂ρhµν = −16πTµν “Lorentz gauge”

8.2 Newtonian limit

Newtonian gravity: ~∇2Φ = 4πρ ; c = G = 1 , Φ ∼ v2 ≪ 1 , ǫ :=v2

c2= v2

⇒ matter sources weak: ρ ∼ O(ǫ)

energy momentum tensor

for Newtonian matter: T00 = ρ+O(ǫ2)

T0i ∼ T00 vi ∼ O(ǫ3/2)

Tij ∼ T00 vivj ∼ O(ǫ2)


E.g. perfect fluid: Tµν = (ρ+ P )uµuν + Pgµν

P ∼ ρv2

c2,P

ρ≈ 10−5 in sun

In SR: uµ =

[1√

1− v2,

vi√1− v2

]

; v2 = vivi

In Newt. gravity temporal changes in Φ are caused by motion of sources

⇒ ∂

∂t∼ v

∂

∂xi= O(ǫ1/2) ∂

∂xi, i = 1 . . . 3

⇒ hµν = ∂ρ∂ρhµν = ∂i∂ihµν = ~∇2hµν = −16πTµν

⇒ ~∇2h00 = −16πT00 = −16πρ+O(ǫ2) , h0i = O(ǫ3/2) , hij = O(ǫ2)

Newton’s law with h00 = −4Φ

⇒ h = ηµν hµν = 4Φ +O(ǫ2) = −h

⇒ h00 = h00 −1

2η00h = −2Φ , hij = hij −

1

2ηij h = −2Φδij

or ds2 = −(1 + 2Φ)dt2 + (1− 2Φ)(dx2 + dy2 + dz2) cf. Sec. 1.4

Geodesics in the weak field

Lagrangian: L = −gµνdxµ

dτ

dxν

dτ

(= G2 in Sec. 3.3

)

= (1 + 2Φ)t2 − δij(1− 2Φ)xixj!= 1 (proper time)

⇒ t2 = (1 + 2Φ)−1[1 + δij x

ixj +O(ǫ2)]

⇒ t = 1− Φ+1

2δij x

ixj +O(ǫ2)

EL-eq. for xk:d

dτ

[−2δjk(1−2Φ)xj

]=

∂L

∂xk= 2∂kΦt

2 + 2∂kΦδij xixj +O(ǫ2)

= 2∂kΦ +O(ǫ2)

⇒ −2δjkxj +O(ǫ2) = 2∂kΦ

⇒ d2xk

dτ 2=d2xk

dt2= −∂kΦ test body in Newt. gravity


8.3 Gravitational waves

weak field but now: vacuum; no longer “∂t ≪ ∂x”

⇒ hµν = (∂2t − ~∇2)hµν = 0

Plane wave solution: hµν = Hµνeikρxρ ; Hµν = const

(i) hµν = 0 ⇒ kµkµ = 0 → speed of light

(ii) Lorentz gauge: ∂ν hµν = 0 ⇒ kµHµν = 0 “transverse”

E.g. wave in z-dir.: kµ = ω (1, 0, 0, 1) ⇒ Hµ0 +Hµ3 = 0

Remaining gauge freedom: take ξµ = Xµeikρxρ ⇒ ∂ν∂νξµ = 0

⇒ . . .⇒ Hµν → Hµν + i(kµXν + kνXµ − ηµνkρXρ)

⇒ . . .⇒ ∃Xµ : H0µ = 0 , Hµµ = 0 “traceless”

In this gauge: 1) h = 0⇒ hµν = hµν

2) plane wave in z-dir.: H0µ = H3µ = Hµµ = 0

⇒ Hµν =

0 0 0 00 H+ H× 00 H× −H+ 00 0 0 0

Effect on particles

Consider particle at rest in background Lorentz frame: uα0 = (1, 0, 0, 0)

geodesic eq.:d

dτuα + Γαµνu

µuν = uα + Γα00 = 0

Γα00 =1

2ηαβ(∂0hβ0 + ∂0h0β − ∂βh00) = 0 since H0µ = 0

⇒ uα = (1, 0, 0, 0) always

⇒ particle stays at xµ = const in this gauge


Proper separation: ds2 = −dt2 + (1 + h+)dx2 + (1− h+)dy2 + 2h×dx dy + dz2

Case 1: H× = 0 , H+ 6= 0 ⇒ h+ oscillates

2 particles at (−δ, 0, 0), (δ, 0, 0) ⇒ ds2 = (1 + h+) 4δ2

2 particles at (0, −δ, 0), (0, δ, 0) ⇒ ds2 = (1− h+) 4δ2

Case 2: H+ = 0 , H× 6= 0

2 particles at (−δ, −δ, 0) /√2 , (δ, δ, 0) /

√2 ⇒ ds2 = (1 + h×) 4δ

2

2 particles at (δ, −δ, 0) /√2 , (−δ, δ, 0) /

√2 ⇒ ds2 = (1− h×) 4δ2

8.4 The field far from a source

weak field with matter: ∂ρ∂ρhµν = −16πTµν

Green’s function: hµν(t, ~x) = 4

∫Tµν(t− |~x− ~y|, ~y)

|~x− ~y| d3y , |~x|2 = x21 + x22 + x23

Assume matter has compact support inside radius d

⇒ far from the source: r := |~x| ≫ d ≥ |~y|

⇒ . . .⇒ |~x− ~y| = r − ~x · ~y +O(dr

); ~x :=

~x

r

⇒ Tµν(t− |~x− ~y|, ~y

)= Tµν(t− r, ~y)− ~x · ~y

(∂0Tµν

)(t− r, ~y) Taylor

Assume v ≪ c ⇒ ∂0Tµν ∼ Tµνv

d≪ Tµν

d

⇒ hµν(t, ~x) ≈4

r

∫

Tµν(t− r, ~y) d3y (∗)


Lorentz gauge: ∂ν hµν = 0

⇒ ∂0h0i = ∂j hji , ∂0h00 = ∂ihoi ; sum over i, j = 1 . . . 3

⇒ Strategy: calculate hij , → h0i → h00

∫

T ijd3y =

∫

∂k(Tik yj)

︸︷︷︸

surface term →0

−(∂kT ik) yj d3y

=

∫

(∂0Ti0) yj d3y since ∂µT

iµ = 0

⇒∫

T (ij)d3y = ∂0

∫

T 0(iyj) d3y = ∂0

∫1

2∂k(T

0kyi yj)︸︷︷︸

→0

−12(∂kT

0k) yi yj d3y

=1

2∂0∂0

∫

T 00yi yj d3y∣∣∣ ∂µT

0µ = 0

⇒ hij(t, ~x) =2

rIij(t− r) ; Iij(t− r) =

∫

T00(t− r, ~y) yi yj d3y

Iij = “Quadrupole tensor”

Next: h0i

∂0h0i = ∂j hji = ∂j

(2

rIij(t− r)

) ∣∣∣ ∂jr =

xj

r

⇒ h0i = ∂j

(2

rIij(t− r)

)

+ Ci = −2xjr2Iij

︸︷︷︸

O( 1r2)→0

−2 xjrIij + Ci

Const. of integration: (∗)⇒ Ci =4

r

∫

T0i(0, ~y) d3y =: −4

rPi “Momentum”

∂0Pi(t− r) = −∫

∂0T0i d3y = −

∫

∂jTji d3y

︸︷︷︸

surface term

= 0

⇒ Pi conserved at leading order

Pi = Ci = 0 in ctr. of mass frame


Finally: h00

∂0h00 = ∂ih0i

⇒ h00 = ∂i

(

−2 xjrIij(t− r)

)

+ C0 =2xixjr

Iij(t− r) + C0+O( 1r2)

C0 =4

r

∫

T00(0, ~y) d3y =:

4

rE “Energy”

∂0E(t− r) = ∂0

∫

T00 d3y =

∫

(∂iTi0) d3y

︸︷︷︸

surface term

= 0

⇒ E conserved at 1st order

At higher order: E, Pi not conserved!

8.5 Energy in gravitational waves

Consider 2nd order pert. theory, vacuum

Notation: gµν = gµν + δ(1)gµν + δ(2)gµν = ηµν + hµν + h(2)µν

E.g.: gµν = ηµν + δ(1)gµν + δ(2)gµν

⇒ gµρgρν = δµν +[hµν + δ(1)gµρ ηρν

]

︸︷︷︸

∼ǫ

+[δ(2)gµρ ηρν + h(2)µν + δ(1)gµρ hρν

]

︸︷︷︸

∼ǫ2

with h(2)µν = ηµρh(2)ρν

⇒ δ(1)gµν = −hµρ =: g(1)µρ[h]

δ(2)gµν = −h(2) µρ + hµσhσν =: g(1)µν [h(2)]

︸︷︷︸

linear in h(2)

+ g(2)µν [h]︸︷︷︸

quadratic in h

Generic pattern in pert. theory: δ(1)Sµν = S(1)µν [h]

δ(2)Sµν = S(1)µν [h

(2)]

+ S(2)µν [h]


Einstein equations

Gµν = Gµν + δ(1)Gµν + δ(2)Gµν

= 0 +G(1)µν [h] +G(1)

µν [h(2)] +G(2)

µν [h]

G(2)µν [h] = R(2)

µν [h]−1

2R(1)[h] hµν −

1

2R(2)[h] ηµν

In vacuum: G(1)µν [h] = R(1)

µν [h] = 0 as before

G(1)µν [h

(2)] = 8πtµν [h]

tµν = −1

8πG(2)µν [h] = −

1

8π

(R(2)µν [h]−

1

2ηρσR(2)

ρσ [h] ηµν)

Contracted Bianchi Identities: gµρ∇ρGµν = 0

at ǫ: ∂µG(1)µν [h] = 0

!⇒ ∂µG(1)µν [h(2)] = 0

∣∣∣ Bianchi Ids. true for ηµν + h

(2)µν !

at ǫ2: Einstein eqs.: Gµν = 0 , δ(1)Gµν = 0

⇒ . . .⇒ ∂µ(G

(2)µν [h]

)= 0

⇒ ∂µtµν = 0 ; like energy-momentum tensor!

⇒ regard tµν as energy momentum of grav. field.

Problem: tµν gauge dependent

“global solution”: integrate over all space → ADM mass...

“local approximation”: use “large” 4-volume V ∼ a4 as follows:

Def.: “average” 〈Xµν〉 :=∫

V

Xµν(x)W (x) d4x

weight W (x) ≥ 0 ,

∫

V

W d4x = 1 , W (x)→ 0 on ∂V smoothly

⇒ 〈∂ρXµν〉 =∫

V

(∂ρXµν)W d4x = −∫

V

Xµν(∂ρW ) d4x


Let Xµν oscillate with wavelength λ ⇒ ∂ρXµν ∼Xµν

λ

Also: ∂ρW ∼W

a, a≫ λ

⇒ 〈∂ρXµν〉 ∼Xµν

a≪ Xµν

λ∼ ∂ρXµν

⇒ neglect total derivs. in 〈 . 〉

⇒ “ 〈A∂B〉 = 〈∂(AB)〉 − 〈(∂A)B〉 ≈ −〈(∂A)B〉 ”

⇒ . . .⇒ (i) 〈ηµνR(2)µν [h]〉 = 0

(ii) 〈tµν〉=1

32π〈∂µhρσ ∂νhρσ −

1

2∂µh ∂νh− 2∂σh

ρσ ∂(µhν)ρ〉

(iii) 〈tµν〉 is gauge invariant

8.6 The quadrupole formula

Energy flux in gravitational waves: −〈t0i〉

consider sphere far from source: r ≫ d ; xi =xi

r

⇒ power 〈p〉 = −∫

r2〈t0i〉xi dΩ ; dΩ := sin θ dθ dφ

Lorentz gauge: ∂ν hνµ = 0

⇒ 〈t0i〉 =1

32π

⟨

∂0hρσ ∂ihρσ − 1

2∂0h ∂ih

⟩

=1

32π

⟨

∂0hjk ∂ihjk − 2∂0h0j ∂ih0j + ∂0h00 ∂ih00 −1

2∂0h ∂ih

⟩

1© 2© 3© 4©

Take hρσ from Sec. 8.4, order O(1/r), do some δij algebra (cf. [5])

⇒ . . .⇒ 〈p〉t =1

5

⟨...Qij

...Qij

⟩

t−r; Qij := Iij −

1

3Ikk δij

valid for: wave zone r ≫ d , λ≫ d (⇔ v ≪ c)

9 DIFFERENTIAL FORMS 72

Examples:

1) binary M1 =M2 =M ⇒ . . .⇒ 〈p〉 ∼(M

d

)5

→ black holes, neutron stars

2) hij ∼M2

dr∼ O(10−21) when the signal reaches the earth

9 Differential forms

Consider curve λ(t), vectord

dt, 1-form ω

⇒∫

λ

ω :=

∫

λ

ω( d

dt

)

dt =

∫

λ

ωµ dxµ

Goal: generalize to areas, . . .

Note: in 3 dims. ~V × ~W is an antisymmetric area element

9.1 p-forms

Def.: “p-form” := totally antisymmetric(0p

)tensor

0-form: function , 1-form: covector

Def.: Let η be a p-form, ω a q-form

(η ∧ ω)a1...apb1...bq :=(p+ q)!

p! q!η[a1...apωb1...bq ]

⇔ η ∧ ω =(p+ q)!

p! q!A[η ⊗ ω]

↑totally antisymm. operator

e.g. ηa ∧ ωb = ηaωb − ηbωa

one can show: 1) η ∧ ω = (−1)p qω ∧ η ; η ∧ η = 0 if p odd

2) (η ∧ ω) ∧ χ = η ∧ (ω ∧ χ)


Basis: Dual basis fµ ⇒ The set of p-forms

fµ1 ∧ . . . ∧ f

µp = p!(f[µ1 ⊗ . . .⊗ f

µp])is a basis for p-forms:

η =1

p!ηµ1...µpf

µ1 ∧ . . . ∧ fµp

Def.: “Exterior derivative” of p-form η := p+ 1 form

(dη)µ1...µp+1 = (p+ 1) ∂[µ1ηµ2...µp+1]

= (p+ 1)[

∇[µ1ηµ2...µp+1] + Γρ[µ2µ1︸︷︷︸

torsion=0

η|ρ|µ3...µp+1] + . . .]

⇒ . . .⇒ 1) d(dη) = 0

2) d(η ∧ ω) = (dη) ∧ ω + (−1)pη ∧ dω

3) d(φ∗η) = φ∗ dη “d , pullback commute”

Def.: A p-form η is “closed” :⇔ dη = 0 .

η is “exact” :⇔ ∃(p− 1) form ω : η = dω .

η exact ⇒ η closed

Poincare Lemma: η closed

⇒ ∀ points r∈M ∃ neighbourhood O of r, (p−1) form ω : η = dω in O


9.2 Integration on manifolds

Lemma: Let ω be a n-form, fµ basis, N n-dim. manifold

⇒ ∃ func. h : ω = h f1 ∧ . . . ∧ fn

Def.: “Orientation” of n-dim. manifold N

:= a smooth nowhere vanishing n-form η.

2 orientations η, η′ are “equivalent” :⇔ ∃ func. h > 0 : η′ = hη

Def.: a coord. chart xµ on N is “right-handed” (RH) relative to orientation η

:⇔ ∃h>0 η = h dx1 ∧ . . . ∧ dxn

Def.: “volume form” on N : ǫ :=√

|g| f1 ∧ . . . ∧ fn ; g := det gµν

Def.: Let ψ = xµ : O ⊂ N → Rn be a RH coord. chart, ω a n-form

∫

O

ω :=

∫

ψ(O)⊂Rn

ω1...n dx1 . . . dxn

can be shown to chart independent

> 1 chart → add patches Oα

Example: scalar f :

∫

O

f ǫ =

∫

ψ(O)

f√

|g| dx1 . . . dxn

Def.: a diffeomorphism φ : N → N is “orientation preserving”

:⇔ φ∗(η) is equivalent to η ∀ orientations η

⇒ . . .⇒∫

N

φ∗(ω) =

∫

N

ω

9.3 Submanifolds, Stokes’ theorem

LetM, N be orientable manifolds of dim. m < n

Def.: “embedding”: φ :M→N , φ smooth, 1-to-1 and

∀p∈M ∃nbhhd. O : φ−1 : φ[O]→M is smooth.

m = n− 1 ⇒ φ[M] is a “hypersurface”


Def.: Let φ[M] be m-dim., η a m-form on N

⇒∫

φ[M]

η :=

∫

M

φ∗(η) ; η = dω ⇒∫

φ[M]

dω =

∫

M

d(φ∗ω) (∗)

Def.: 12Rn :=

(x1, . . . , xn) ∈ R

n∣∣ x1 ≤ 0

N = “manifold with boundary”: like manifold, but charts ψα : Oα → 12Rn

“boundary” := ∂N :=p ∈ N

∣∣ x1(p) = 0

is n− 1 dim.

(x2, . . . , xn) is right-handed on ∂N :⇔ (x1, . . . , xn) is RH on N

Stokes’ Theorem:

For a n-dim. orientable mfld. N with boundary ∂N and (n− 1)-form η

∫

N

dη =

∫

∂N

η

where the rhs. is defined through (∗) with φ : ∂N → N , p 7→ p (∗∗)

Def.: a) X ∈ Tp(N ) is “tangent to φ[M]

:⇔ ∃ curve in φ[M] with tangent X

b) n ∈ T ∗p (N ) is “normal” to φ[M]

:⇔ n(X) = 0 ∀X tangent to φ[M]

Def.: Let Σ be a hypersurface of a Lorentzian mfld., n its normal field.

Σ is “timelike” (“spacelike”, “null”) :⇔ n is spacelike (timelike, null)

On ∂N : x1 = 0 ⇒ dx1 is outgoing normal to ∂N

⇒ n =dx1

√

±g(dx1, dx1)= unit normal

Divergence Theorem:

Let ∂N be time or spacelike, X a VF on N , hµν := φ∗gµν , φ as in (∗∗)

⇒∫

N

∇aXa√

|g| dnx =

∫

∂N

naXa√

|h| dn−1x

10 THE INITIAL VALUE PROBLEM 76

10 The initial value problem

10.1 Extrinsic curvature

Let N be a manifold, Σ a hypersurface, g the metric (Riemannian or Lorentzian)

Unit normal to σ : nana = ∓1 ; upper sign: n timelike

lower sign: n spacelike

Def.: “Projector” ⊥ab := δab ± nanbProjection of tensor: ⊥T ab...cd... = ⊥ae⊥bf . . .⊥gc⊥hd . . . T ef...gh...⇒ 1) ⊥abnb = 0 , ⊥ac⊥cb = ⊥ab

2) ∀X ∈ Tp(N ) : ⊥abXb tangent to Σ , Xa = ⊥abXb ∓ nanbXb

3) X, Y tangent to Σ ⇒ gabXaY b = ⊥abXaY b

⇒ ⊥ab = induced metric on Σ ,

We write γab = ⊥ab “1st fundamental form”

Let X, Y be tangent VFs to Σ, N normal VF

par. transport N along int. curve of X : Xb∇bNa = 0

Does N remain normal to Σ ? No!

Xb∇b(YaNa) = NaX

b∇bYa

N

Y

Y

N

Def.: Extend unit normal n in nbhd. of Σ with nana = ∓1

“extrinsic curvature” := K : Tp(N )× Tp(N )→ R , X , Y 7→ na(∇⊥X(⊥Y )

)a

Note: sign convention

Lemma: Kab = −⊥ca⊥db∇cnd indep. of extension


Proof: 1) KabXaY b = na(⊥X)c∇c(⊥Y )a = −(⊥X)c(⊥Y )a∇cna

∣∣∣ na(⊥Y )a = 0

= −⊥cbXb⊥adY d∇cna

⇒ Kbd = −⊥cb⊥ad∇cna

2) n′a another extension → ma = n′

a − na = 0 on Σ

⇒On Σ : XaY b(Kab−K ′ab) = ⊥ca⊥dbXaY b∇cmd

= (⊥X)c[(⊥Y )d∇cmd + md

︸︷︷︸

=0

∇c(⊥Y )d]

= (⊥X)c∇c

(md(⊥Y )d

)= 0

∣∣∣ deriv. inside Σ

Comment: nb∇cnb =1

2∇c(nbn

b) = 0

⇒ Kab = −⊥ca⊥db∇cnd = −⊥ca(δdb ± ndnb)∇cnd = −⊥ca∇cnb

Def.: Let t : N → R with t = const and normal dt 6= 0 on Σ

⇒ unit normal n = ∓αdt , α := 1√

∓g−1(dt, dt)= “Lapse function”

↑n future pointing if timelike

Lemma: Kab = Kba

Proof: ∇cnd = ∓∇c(α dtd) = ∓α∇c∇dt + (∇cα)ndα

⇒ Kab = +⊥ca⊥db α∇c∇dt + 0 is symmetric (torsion = 0)

Def.: K := Kbb = gabKab


10.2 The Gauss-Codazzi equations

Def.: Covariant deriv. Da on Σ:

DaTb1b2...

c1c2... := ⊥da⊥b1e1⊥b2e2 . . .⊥f1c1⊥f2c2 . . .∇dTe1e2...

f1f2...

⇒ . . .⇒D is torsion free and Levi-Civita conn. of γab on Σ if ∇ is that of gab on N

Daγbc = 0

D defines the Riemann tensor of γab: Rabcd

One can calculate the projections of Rabcd from the Ricci Identity:

Gauss eq.:

Contracted Gauss:

Scalar Gauss:

Codazzi eq.:

Contr. Codazzi:

⊥Rabcd = Ra

bcd ± 2Ka[cKd]b

⊥Rab±⊥cand⊥ebnfRcdef = Rab±KKab∓KacKcb

R± 2Rcd ncnd = R±K2 ∓KcdK

cd

⊥da⊥eb⊥f cngRdefg = −DaKbc +DbKac

⊥cbRcd nd = −DaK

ab +DbK

10.3 The constraint equations

From now on n timelike, “upper sign”

Project Einstein eqs.: Gab = 8πTab

1) EM tensor: ρ := Tabnanb , ja := −⊥baTbcnc , Sab := ⊥Tab

⇒ Tab = ρnanb + janb + jbna + Sab , T = T bb = −ρ+ S

2) n-n proj.: Rabnanb +

1

2R = 8πρ

∣∣∣ ← scalar Gauss

⇒ R−KcdKcd +K2 − 16πρ = 0 “Hamiltonian constraint”

3) n-⊥ proj.: ⊥banc(

Rbc −1

2gbcR

)

= ⊥bancRbc = −8πja∣∣∣ ← contr. Codazzi

⇒ DcKca −DaK − 8πja = 0 “momentum constraint”


10.4 Foliations

Def.: “Cauchy surface” := spacelike hypersurface Σ in N such that

each timelike or null curve without endpoint intersects Σ exactly once.

(N , g) is “globally hyperbolic” :⇔ it admits a Cauchy surface

From now on: Let (N , g) be globally hyperbolic.

⇒ . . .⇒ ∃ smooth t : N → R , dt 6= 0 everywhere and hypersurfaces Σ are level

surfaces t = const : ∀t∈R Σt =p ∈ N : t(p) = t

, Σt ∩ Σt′ = ∅ ⇔ t 6= t′

We assume: Σt spacelike, N = ∪t∈R

Σt ; this is called a “foliation” of N .

From now on: use just t (no t)

Def.: m := αn “normal evolution vector”

Note: n = −αdt , n · n = −1

⇒ m ·m = −α2 , 〈dt,m〉 = − 1

α〈n,m〉 = −〈n,n〉 = 1

⇒ Lmt = m(t) = 〈dt,m〉 = 1

⇒ Proper time along int. curve of m (cf. Sec. 3.2) :

τ =

∫ t

t0

√

−g(m,m) dt ⇒ dτ

dt=√

−g(m,m) = α

Def.: “acceleration” ab := nc∇cnb

Lemma: ab = Db lnα

Recall: Kab = −⊥ca∇cnb = −∇anb − nanc∇cnb

⇒ ∇anb = −Kab − naab = −Kab − naDb lnα

⇒ ∇amb = ∇a(α nb) = nb∇aα + α∇anb

⇒ ∇amb = nb∇aα− αKab − naDbα


Lemma: 1) Lmγab = −2αKab

2) Lnγab = −2Kab

3) Lmγab = Lm⊥ab = 0

4) Lnγab = naDb lnα

Corollary: Let T be a tangent tensor: ⊥T = T

⇒ LmT = Lm(⊥T ) = (Lm⊥)T +⊥LmT∣∣∣ Lm⊥ab = Lmγ

ab = 0

⇒ LmT is tangent to Σ

With these tools we can calculate the final projection: ⊥eanf⊥gbnhRefgh

Starting point: Ricci Identity; cf. Sec.3.4.1 in [2]

⇒ . . .⇒ ⊥ea⊥gbnhRefghnf =

1

αLmKab +KacK

cb +

1

αDaDbα

with contracted Gauss eq.: ⊥Rab = −1

αLmKab −

1

αDaDbα +Rab +KKab − 2KacK

cb (∗)

·⊥ab , use scalar Gauss: R =2

αLmK −

2

αDcD

cα +R+K2 +KcdKcd

10.5 The 3+1 equations

Einstein eqs.: Rab −1

2gabR = 8πTab ⇒ − R = 8πT

⇒ Rab = 8π

(

Tab −1

2gabT

) ∣∣∣ ⊥·

⇒ ⊥Rab = 4π(2Sab + (ρ− S)γab

)

(∗)⇒ LmKab = −DaDbα + α

Rab +KKab − 2KacKcb + 4π

[(S − ρ)− 2Sab

]

Open question: Relate Lm to a time derivative∂

∂t


Adapted coordinates: xα = (t, xi) , i = 1, 2, 3 , xi label points in Σt

→ basis ∂t , ∂i ; dual basis dt , dxi

Integral curves of the ∂i have t = const, i.e. are in Σt

What about ∂t? Clearly 〈dt,∂t〉 = 1 = 〈dt,m〉 ⇒ 〈dt,∂t −m〉 = 0

Def.: “shift vector” β := ∂t −m ⇒ 〈dt,β〉 = 0

⇒ ∂t = αn+ β

Curves xi = const are in general not normal to Σt.

β measures this deviation.

∂tn

m

β

Metric components: g00 = g(∂t,∂t) = . . . = −α2 + β · β etc.

⇒ . . .⇒ gαβ =

(−α2 + βkβ

k βjβi γij

)

⇔ gαβ =

(−α−2 α−2βj

α−2βi γij − α−2βiβj

)

det gαβ = −α2 det γij ⇒√−g = α

√γ

In adapted coords.: The 3+1 eqs. contain only tensors tangent to Σt

⇒ we can ignore time components

⇒ substitute i, j, . . . = 1, 2, 3 for abstract indices

We have: Lmγij = L∂tγij − Lβγij =

∂

∂tγij − βm∂mγij − γmj∂iβm − γim∂jβm

LmKij =∂

∂tKij − βm∂mKij −Kmj∂iβ

m −Kim∂jβm

⇒ ∂tγij = Lβγij − 2αKij

∂tKij = LβKij −DiDjα + α

Rij +KKij − 2KimKmj + 4π

[(S − ρ)γij − 2Sij

]

R+K2 −KmnKmn − 16πρ = 0

DmKmi −DiK − 8πji = 0

Comments: 1) α, βi freely specifiable! → gauge freedom

2) Bianchi Identities ⇒ . . .⇒ constraints preserved under evolution

3) numerical relativity → need new variables

11 THE LAGRANGIAN FORMULATION 82

11 The Lagrangian formulation

Consider scalar field in curved spacetime: S =

∫

M

[

−12gαβ∇αΦ∇βΦ− V (Φ)

]√−g d4x

Vary with respect to Φ; assume δΦ vanishes on ∂M; use divergence theorem

⇒ δS = S[Φ + δΦ]− S[Φ]

=

∫

M

(− gαβ∇αΦ∇βδΦ− V ′(Φ) δΦ

)√−g d4x∫

M

[−∇α(δΦ∇αΦ) + δΦ∇α∇αΦ− V ′(Φ)δΦ

]√−g d4x∫

∂M

−δΦ nα∇αΦ√

|h|d3x︸︷︷︸

=0

+

∫

M

(∇α∇αΦ− V ′(Φ)

)δΦ√−g d4x

⇒∇α∇αΦ− V ′(Φ) = 0 “Eqs. of motion”

Goal: same for GR

Sign convention: 1) Unit normal: n always outward; past, future, spacelike!

2) Extrinsic curvature: Kab = +⊥∇anb

Saves us case distinctions on spacelike boundaries.

n

n

nV

∂V

The action in GR: SGR[g, φ] =1

16π

(IH [g] + IB[g]− I0

)+ SM [φ, g]

1) Hilbert term: IH =

∫

V

R√−g d4x

2) boundary term: IB = 2

∮

∂V

K√

|γ|d3y

3) constant term: I0 = 2

∮

∂V

K0

√

|γ| d3y

4) matter term: SM =

∫

V

L(φ, φ,α; gαβ)√−g d4x

It is convenient to vary gαβ instead of gαβ: gαµgµβ = δαβ ⇒ δgαβ = −gαµgβνδgµν

Lemma: δ√−g = −1

2

√−ggαβ δgαβ


1) δIH =

∫

V

δ(gαβRαβ

√−g)d4x

=

∫

V

Rαβ

√−g δgαβ + gαβ√−g δRαβ +Rδ

√−g d4x

=

∫

V

(

Rαβ −1

2Rgαβ

)

︸︷︷︸

Einstein eqs.

δgαβ√−g d4x+

∫

V

gαβ δRαβ

√−g d4x︸︷︷︸

?

In normal coords.: δRαβ∗= δ(Γµαβ,µ − Γµαµ,β

)∣∣∣ Γ = 0

∗= δΓµαβ,µ − δΓµαµ,β∗= δΓµαβ;µ − δΓµαµ;β

∣∣∣ Γ = 0 , δΓ = tensor!

tensorial eq. ⇒ valid in any coords.!

⇒∫

V

gαβ δRαβ

√−g d4x=∫

Xµ;µ

√−g d4x ; Xµ := gαβ δΓµαβ − gαµ δΓβαβ

=

∮

∂V

Xµnµ√

|γ| d3y∣∣∣ Divergence theorem

On ∂V : δgαβ = 0 = δgαβ

⇒ δΓµαβ =1

2gµν(δgνα,β + δgνβ,α − δgαβ,ν

)

⇒ . . .⇒ Xµ = gµν gαβ(δgνα,β − δgαβ,ν

)

︸︷︷︸

=Xν

⇒ nµXµ = nµ(γαβ ∓ nαnβ)(δgµβ,α − δgαβ,µ︸︷︷︸

antisymm. in α, µ

)

= nµ γαβ(δgµβ,α

︸︷︷︸−δgαβ,µ

)

= deriv. of δgµβ tangent to ∂V → 0

⇒ δIH =

∫

V

Gαβ δgαβ√−g d4x−

∮

∂V

γαβ δgαβ,µ nµ√

|γ| d3y (∗)


2) K = γαβKαβ = γαβ∇αnβ = γαβ(∂αnβ − Γµαβnµ)

⇒ δK = −γαβδΓµαβ nµ = −γαβ δΓµαβnµ

= −12γαβ(δgµα,β + δgµβ,α − δgαβ,µ

)nµ

=1

2γαβ δgαβ,µn

µ∣∣∣ tang. derivs of gαβ vanish on ∂V

⇒ δIB =

∮

∂V

γαβ δgαβ,µnµ|γ|1/2 d3y cancels term in (∗)

3) I0 depends on gαβ only through√

|γ|

⇒ δI0 = 0 on ∂V

⇒ no effect on eqs. of motion, but on numerical value of SGR

Let gαβ be a solution of the vacuum eqs. Rαβ = 0 ⇒ R = 0

⇒ SGR +1

16πI0 =

1

16πIB =

1

8π

∮

K |γ|1/2 d3y

evaluate on closed 3-cylinder for a flat spacetime:

on Σt1 , Σt2 : K = 0

at r = R : K = nα;α = . . . =2

R, |γ|1/2 = R2 sin θ

⇒∮

∂V

K |γ|1/2 d3y = 8πR(t2 − t1) diverges as R→∞

This divergence persists in curved spacetimes

⇒ cured by I0 with K0 = curvature of ∂V embedded in flat spacetime

Σt2

Σt1

σR

r = R

4) δSM =

∫

V

∂L

∂gαβδgαβ

√−g + L δ√−g d4x =

∫

V

(∂L

∂gαβ− 1

2Lgαβ

)

δgαβ√−g d4x

Def.: Tαβ := −2 ∂L

∂gαβ+ Lgαβ Energy-momentum tensor

⇒ δSM = −12

∫

V

Tαβ δgαβ√−g d4x

Conclusion: δ

[1

16π

(IH + IB − I0

)+ SM

]

= 0 ⇒ Gαβ = 8πTαβ

REFERENCES 85

References

[1] S. M. Carroll. Lecture notes on general relativity, 1997. gr-qc/9712019.

[2] E. Gourgoulhon. 3+1 Formalism and Bases of Numerical Relativity. 2007. gr-qc/0703035.

[3] S. W. Hawking and G. F. R. Ellis. The Large Scale Structure of Space-Time. CambridgeUniversity Press, 1973.

[4] J. Stewart. Advanced general relativity. Cambridge University Press, 1991.

[5] Harvey Reall’s lecture notes on General Relativity:http://www.damtp.cam.ac.uk/user/hsr1000/teaching.html.

[6] Eric Poisson’s Lecture Notes:http://www.physics.uoguelph.ca/poisson/research/notes.html.

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